N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method.

Example 1

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 1)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= -2 \, t y + 4 \, x y - 1\hspace{2em}x( -2 )= -2\]

\[y'= -2 \, x^{2} y + t y^{2} + 1\hspace{2em}y( -2 )= -2\]

Answer.

\(x(-1.8)\approx -0.6000\) and \(y(-1.8)\approx -0.2000\).

\(x(-1.6)\approx -0.8480\) and \(y(-1.6)\approx 0.01440\).

\(x(-1.4)\approx -1.049\) and \(y(-1.4)\approx 0.2102\).

Example 2

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 2)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -4 \, t^{2} x - t^{2} y - 3\hspace{2em}x( -1 )= 1\]

\[y'= t^{2} y^{2} + t x\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.80)\approx 0.0000\) and \(y(-0.80)\approx -1.400\).

\(x(-0.60)\approx -0.4208\) and \(y(-0.60)\approx -1.149\).

\(x(-0.40)\approx -0.8169\) and \(y(-0.40)\approx -1.004\).

Example 3

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 3)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= -4 \, t^{2} x^{2} + 4 \, t^{2} y - 1\hspace{2em}x( -2 )= 0\]

\[y'= -2 \, t^{2} x + 2 \, t y - 3\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.8)\approx -0.2000\) and \(y(-1.8)\approx -0.6000\).

\(x(-1.6)\approx -2.059\) and \(y(-1.6)\approx -0.5088\).

\(x(-1.4)\approx -11.98\) and \(y(-1.4)\approx 1.325\).

Example 4

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 4)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} - 3 \, t x + 1\hspace{2em}x( 0 )= 2\]

\[y'= -t^{2} x - 2 \, t y - 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.20)\approx 2.200\) and \(y(0.20)\approx -0.2000\).

\(x(0.40)\approx 1.981\) and \(y(0.40)\approx -0.4016\).

\(x(0.60)\approx 1.199\) and \(y(0.60)\approx -0.6007\).

Example 5

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 5)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= t x - 3 \, x y\hspace{2em}x( 0 )= 2\]

\[y'= t^{2} x - 4 \, t^{2} y + 1\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.20)\approx 0.8000\) and \(y(0.20)\approx 1.200\).

\(x(0.40)\approx 0.2560\) and \(y(0.40)\approx 1.368\).

\(x(0.60)\approx 0.06635\) and \(y(0.60)\approx 1.401\).

Example 6

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 6)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

\[x'= -4 \, t^{2} y^{2} + t x + 3\hspace{2em}x( 2 )= -2\]

\[y'= t^{2} x - 2 \, x y - 1\hspace{2em}y( 2 )= 2\]

Answer.

\(x(2.2)\approx -15.00\) and \(y(2.2)\approx 1.800\).

\(x(2.4)\approx -33.55\) and \(y(2.4)\approx -2.120\).

\(x(2.6)\approx -69.76\) and \(y(2.6)\approx -69.41\).

Example 7

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 7)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x^{2} + 3 \, t y\hspace{2em}x( 0 )= -1\]

\[y'= -4 \, t^{2} x + 2 \, t y^{2} - 3\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx -1.000\) and \(y(0.10)\approx -1.300\).

\(x(0.20)\approx -1.035\) and \(y(0.20)\approx -1.562\).

\(x(0.30)\approx -1.112\) and \(y(0.30)\approx -1.748\).

Example 8

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 8)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x + 4 \, x y - 1\hspace{2em}x( 1 )= -2\]

\[y'= 2 \, t^{2} x + t y - 3\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.1)\approx -2.900\) and \(y(1.1)\approx -0.7000\).

\(x(1.2)\approx -3.592\) and \(y(1.2)\approx -1.779\).

\(x(1.3)\approx -3.205\) and \(y(1.3)\approx -3.327\).

Example 9

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 9)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -3 \, t^{2} x^{2} - x^{2} y^{2} - 1\hspace{2em}x( 0 )= 1\]

\[y'= 4 \, x^{2} y + 3 \, t y + 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.10)\approx 0.5000\) and \(y(0.10)\approx 3.000\).

\(x(0.20)\approx 0.1742\) and \(y(0.20)\approx 3.590\).

\(x(0.30)\approx 0.03475\) and \(y(0.30)\approx 4.049\).

Example 10

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 10)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y - 2 \, t y + 3\hspace{2em}x( 0 )= -2\]

\[y'= 3 \, t^{2} x + 4 \, t^{2} y + 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.20)\approx -1.400\) and \(y(0.20)\approx 0.2000\).

\(x(0.40)\approx -1.130\) and \(y(0.40)\approx 0.3728\).

\(x(0.60)\approx -0.9698\) and \(y(0.60)\approx 0.5121\).

Example 11

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 11)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -t y^{2} - 2 \, x y\hspace{2em}x( -1 )= 1\]

\[y'= 3 \, t^{2} y + 2 \, x^{2} y - 3\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.80)\approx 1.000\) and \(y(-0.80)\approx -0.6000\).

\(x(-0.60)\approx 1.298\) and \(y(-0.60)\approx -1.670\).

\(x(-0.40)\approx 2.499\) and \(y(-0.40)\approx -3.756\).

Example 12

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 12)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y^{2} + 2 \, x^{2} y + 2\hspace{2em}x( 1 )= -2\]

\[y'= -4 \, x^{2} y^{2} + t x^{2} - 2\hspace{2em}y( 1 )= 1\]

Answer.

\(x(1.1)\approx -0.8000\) and \(y(1.1)\approx -0.4000\).

\(x(1.2)\approx -0.6125\) and \(y(1.2)\approx -0.5706\).

\(x(1.3)\approx -0.3615\) and \(y(1.3)\approx -0.7744\).

Example 13

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 13)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= t^{2} y - x^{2} y + 3\hspace{2em}x( -2 )= -2\]

\[y'= x^{2} y^{2} + t y^{2}\hspace{2em}y( -2 )= -2\]

Answer.

\(x(-1.9)\approx -1.700\) and \(y(-1.9)\approx -1.200\).

\(x(-1.8)\approx -1.486\) and \(y(-1.8)\approx -1.057\).

\(x(-1.7)\approx -1.295\) and \(y(-1.7)\approx -1.012\).

Example 14

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 14)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x^{2} y - 4 \, t y^{2} - 1\hspace{2em}x( -1 )= 2\]

\[y'= t x + 4 \, t y + 2\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.90)\approx 3.100\) and \(y(-0.90)\approx 0.6000\).

\(x(-0.80)\approx 4.283\) and \(y(-0.80)\approx 0.3050\).

\(x(-0.70)\approx 5.331\) and \(y(-0.70)\approx 0.06478\).

Example 15

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 15)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 3 \, t x^{2} + t y - 1\hspace{2em}x( -1 )= 2\]

\[y'= -2 \, t x^{2} + 4 \, x^{2} y - 3\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.90)\approx 0.8000\) and \(y(-0.90)\approx -2.100\).

\(x(-0.80)\approx 0.7162\) and \(y(-0.80)\approx -2.822\).

\(x(-0.70)\approx 0.7189\) and \(y(-0.70)\approx -3.619\).

Example 16

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 16)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= x^{2} y^{2} + t y + 2\hspace{2em}x( -1 )= 1\]

\[y'= 2 \, t^{2} y^{2} + 2 \, x y^{2} + 2\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.90)\approx 1.200\) and \(y(-0.90)\approx 0.2000\).

\(x(-0.80)\approx 1.388\) and \(y(-0.80)\approx 0.4161\).

\(x(-0.70)\approx 1.588\) and \(y(-0.70)\approx 0.6863\).

Example 17

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 17)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x^{2} y + 2 \, t y^{2} - 2\hspace{2em}x( -1 )= -2\]

\[y'= t^{2} y^{2} + 3 \, x y + 3\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.90)\approx -2.200\) and \(y(-0.90)\approx 0.3000\).

\(x(-0.80)\approx -2.126\) and \(y(-0.80)\approx 0.4093\).

\(x(-0.70)\approx -1.983\) and \(y(-0.70)\approx 0.4590\).

Example 18

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 18)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y - 4 \, t x\hspace{2em}x( -2 )= 0\]

\[y'= 2 \, t x^{2} - 4 \, x y - 2\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.9)\approx 0.0000\) and \(y(-1.9)\approx -0.2000\).

\(x(-1.8)\approx 0.2166\) and \(y(-1.8)\approx -0.4000\).

\(x(-1.7)\approx 0.7614\) and \(y(-1.7)\approx -0.5822\).

Example 19

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 19)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -2 \, x^{2} y^{2} + 2 \, t x - 3\hspace{2em}x( 0 )= 2\]

\[y'= 3 \, t^{2} y^{2} - x^{2} y^{2} + 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx 0.9000\) and \(y(0.10)\approx -1.300\).

\(x(0.20)\approx 0.3442\) and \(y(0.20)\approx -1.332\).

\(x(0.30)\approx 0.01596\) and \(y(0.30)\approx -1.232\).

Example 20

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 20)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, t^{2} y^{2} - 4 \, x^{2} y^{2} - 3\hspace{2em}x( 0 )= 0\]

\[y'= -t^{2} y + x y^{2} + 1\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.10)\approx -0.3000\) and \(y(0.10)\approx 1.100\).

\(x(0.20)\approx -0.6399\) and \(y(0.20)\approx 1.163\).

\(x(0.30)\approx -1.145\) and \(y(0.30)\approx 1.171\).

Example 21

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 21)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -2 \, t x^{2} + x^{2} y\hspace{2em}x( 1 )= 2\]

\[y'= 2 \, t^{2} x^{2} + 2 \, t^{2} y - 2\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.2)\approx 0.4000\) and \(y(1.2)\approx 1.200\).

\(x(1.4)\approx 0.3616\) and \(y(1.4)\approx 1.583\).

\(x(1.6)\approx 0.3298\) and \(y(1.6)\approx 2.527\).

Example 22

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 22)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -x^{2} y + 4 \, t x + 3\hspace{2em}x( 0 )= 0\]

\[y'= -2 \, t^{2} y^{2} + 3 \, t^{2} x + 2\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.20)\approx 0.6000\) and \(y(0.20)\approx 1.400\).

\(x(0.40)\approx 1.195\) and \(y(0.40)\approx 1.783\).

\(x(0.60)\approx 1.668\) and \(y(0.60)\approx 2.094\).

Example 23

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 23)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -t x^{2} - 4 \, x y + 3\hspace{2em}x( 1 )= 2\]

\[y'= -3 \, t x - 3 \, t y - 3\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.1)\approx 3.500\) and \(y(1.1)\approx -2.300\).

\(x(1.2)\approx 5.672\) and \(y(1.2)\approx -2.996\).

\(x(1.3)\approx 8.909\) and \(y(1.3)\approx -4.260\).

Example 24

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 24)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y^{2} - 4 \, x^{2} y^{2} + 2\hspace{2em}x( 1 )= -1\]

\[y'= -3 \, x^{2} y + 2 \, t y - 3\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.1)\approx -0.8000\) and \(y(1.1)\approx -0.3000\).

\(x(1.2)\approx -0.6013\) and \(y(1.2)\approx -0.6084\).

\(x(1.3)\approx -0.3482\) and \(y(1.3)\approx -0.9884\).

Example 25

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 25)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x y^{2} + 4 \, t y - 1\hspace{2em}x( -1 )= 2\]

\[y'= 2 \, t x^{2} - 2 \, x y^{2} - 3\hspace{2em}y( -1 )= 2\]

Answer.

\(x(-0.90)\approx 2.700\) and \(y(-0.90)\approx -0.7000\).

\(x(-0.80)\approx 3.117\) and \(y(-0.80)\approx -2.577\).

\(x(-0.70)\approx 7.980\) and \(y(-0.70)\approx -8.570\).

Example 26

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 26)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= t^{2} y^{2} - x y - 3\hspace{2em}x( 0 )= 0\]

\[y'= -4 \, x^{2} y^{2} - 4 \, t x + 2\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.20)\approx -0.6000\) and \(y(0.20)\approx 1.400\).

\(x(0.40)\approx -1.016\) and \(y(0.40)\approx 1.332\).

\(x(0.60)\approx -1.289\) and \(y(0.60)\approx 0.5917\).

Example 27

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 27)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= x y^{2} - 4 \, t y + 1\hspace{2em}x( 1 )= -2\]

\[y'= -3 \, t^{2} x^{2} - 4 \, x y + 2\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.1)\approx -1.900\) and \(y(1.1)\approx -1.000\).

\(x(1.2)\approx -1.550\) and \(y(1.2)\approx -2.870\).

\(x(1.3)\approx -1.349\) and \(y(1.3)\approx -5.488\).

Example 28

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 28)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x - 4 \, t y^{2}\hspace{2em}x( -2 )= 0\]

\[y'= -4 \, x^{2} y + 2 \, t y - 3\hspace{2em}y( -2 )= 1\]

Answer.

\(x(-1.9)\approx 0.8000\) and \(y(-1.9)\approx 0.3000\).

\(x(-1.8)\approx 1.446\) and \(y(-1.8)\approx -0.1908\).

\(x(-1.7)\approx 2.409\) and \(y(-1.7)\approx -0.2625\).

Example 29

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 29)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -4 \, t^{2} y^{2} - x y^{2}\hspace{2em}x( 0 )= 2\]

\[y'= 4 \, t^{2} y^{2} + 2 \, x y - 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.10)\approx 1.200\) and \(y(0.10)\approx 2.600\).

\(x(0.20)\approx 0.3618\) and \(y(0.20)\approx 3.051\).

\(x(0.30)\approx -0.1239\) and \(y(0.30)\approx 3.221\).

Example 30

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 30)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, t x^{2} + t y - 2\hspace{2em}x( -1 )= 1\]

\[y'= 4 \, t x - 4 \, t y - 1\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.90)\approx 0.8000\) and \(y(-0.90)\approx -3.300\).

\(x(-0.80)\approx 0.7818\) and \(y(-0.80)\approx -4.876\).

\(x(-0.70)\approx 0.8741\) and \(y(-0.70)\approx -6.786\).

Example 31

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 31)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} + t x + 3\hspace{2em}x( -1 )= 0\]

\[y'= 2 \, x^{2} y^{2} - 2 \, t^{2} y - 3\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.90)\approx 0.3000\) and \(y(-0.90)\approx -0.3000\).

\(x(-0.80)\approx 0.5698\) and \(y(-0.80)\approx -0.5498\).

\(x(-0.70)\approx 0.7849\) and \(y(-0.70)\approx -0.7598\).

Example 32

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 32)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x + 2 \, x y + 1\hspace{2em}x( -2 )= -2\]

\[y'= -2 \, t y^{2} + 2 \, x y - 3\hspace{2em}y( -2 )= -1\]

Answer.

\(x(-1.8)\approx -5.800\) and \(y(-1.8)\approx 0.0000\).

\(x(-1.6)\approx -16.88\) and \(y(-1.6)\approx -0.6000\).

\(x(-1.4)\approx -38.55\) and \(y(-1.4)\approx 3.080\).

Example 33

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 33)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -t^{2} x^{2} + x y^{2} + 2\hspace{2em}x( -2 )= -2\]

\[y'= -4 \, t^{2} y^{2} + t^{2} x + 2\hspace{2em}y( -2 )= 1\]

Answer.

\(x(-1.9)\approx -3.600\) and \(y(-1.9)\approx -1.200\).

\(x(-1.8)\approx -8.597\) and \(y(-1.8)\approx -4.379\).

\(x(-1.7)\approx -48.83\) and \(y(-1.7)\approx -31.82\).

Example 34

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 34)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x^{2} + 4 \, x y - 3\hspace{2em}x( -2 )= 1\]

\[y'= -t y - 2 \, x y + 1\hspace{2em}y( -2 )= -1\]

Answer.

\(x(-1.9)\approx 1.500\) and \(y(-1.9)\approx -0.9000\).

\(x(-1.8)\approx 3.097\) and \(y(-1.8)\approx -0.7010\).

\(x(-1.7)\approx 11.25\) and \(y(-1.7)\approx -0.2930\).

Example 35

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 35)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x^{2} + 3 \, t^{2} y - 1\hspace{2em}x( -1 )= 0\]

\[y'= -4 \, x^{2} y^{2} + 2 \, t y^{2} + 1\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.80)\approx 0.4000\) and \(y(-0.80)\approx 0.8000\).

\(x(-0.60)\approx 0.5482\) and \(y(-0.60)\approx 0.7133\).

\(x(-0.40)\approx 0.5455\) and \(y(-0.40)\approx 0.6689\).

Example 36

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 36)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -2 \, x^{2} y^{2} - 4 \, t^{2} x\hspace{2em}x( -1 )= 2\]

\[y'= 3 \, x y^{2} + 3 \, t y + 2\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.80)\approx 0.4000\) and \(y(-0.80)\approx 0.4000\).

\(x(-0.60)\approx 0.1850\) and \(y(-0.60)\approx 0.6464\).

\(x(-0.40)\approx 0.1260\) and \(y(-0.40)\approx 0.8601\).

Example 37

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 37)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -t^{2} x^{2} + 4 \, t y^{2} - 1\hspace{2em}x( 2 )= 2\]

\[y'= 4 \, t^{2} x - 3 \, t^{2} y\hspace{2em}y( 2 )= 2\]

Answer.

\(x(2.1)\approx 3.500\) and \(y(2.1)\approx 2.800\).

\(x(2.2)\approx 4.583\) and \(y(2.2)\approx 5.270\).

\(x(2.3)\approx 18.75\) and \(y(2.3)\approx 6.492\).

Example 38

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 38)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -t^{2} x^{2} + 3 \, t^{2} y + 1\hspace{2em}x( 1 )= 0\]

\[y'= x^{2} y + t y^{2} + 3\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.2)\approx -1.000\) and \(y(1.2)\approx -0.6000\).

\(x(1.4)\approx -1.606\) and \(y(1.4)\approx -0.03360\).

\(x(1.6)\approx -2.457\) and \(y(1.6)\approx 0.5494\).

Example 39

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 39)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -3 \, t y^{2} - 4 \, t x - 2\hspace{2em}x( 0 )= -1\]

\[y'= -2 \, t^{2} y^{2} - 2 \, t^{2} x + 3\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.20)\approx -1.400\) and \(y(0.20)\approx 2.600\).

\(x(0.40)\approx -2.387\) and \(y(0.40)\approx 3.114\).

\(x(0.60)\approx -4.351\) and \(y(0.60)\approx 3.246\).

Example 40

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 40)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -t^{2} y^{2} + 3 \, x^{2} y + 1\hspace{2em}x( -2 )= 0\]

\[y'= 3 \, t^{2} x^{2} + 4 \, x y\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.9)\approx 0.1000\) and \(y(-1.9)\approx 0.0000\).

\(x(-1.8)\approx 0.2000\) and \(y(-1.8)\approx 0.01083\).

\(x(-1.7)\approx 0.3001\) and \(y(-1.7)\approx 0.05058\).

Example 41

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 41)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x^{2} - 4 \, t^{2} y\hspace{2em}x( -1 )= 1\]

\[y'= t x^{2} - x y^{2} - 3\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.90)\approx 2.000\) and \(y(-0.90)\approx -2.800\).

\(x(-0.80)\approx 3.555\) and \(y(-0.80)\approx -5.028\).

\(x(-0.70)\approx 6.460\) and \(y(-0.70)\approx -15.33\).

Example 42

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 42)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t^{2} y^{2} - 2 \, x y^{2}\hspace{2em}x( 0 )= -2\]

\[y'= -t^{2} y^{2} - 2 \, t^{2} x - 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx -1.600\) and \(y(0.10)\approx -1.100\).

\(x(0.20)\approx -1.208\) and \(y(0.20)\approx -1.198\).

\(x(0.30)\approx -0.8383\) and \(y(0.30)\approx -1.294\).

Example 43

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 43)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

\[x'= t^{2} x + 2 \, t y + 3\hspace{2em}x( 2 )= -2\]

\[y'= -3 \, x^{2} y + 4 \, t y + 3\hspace{2em}y( 2 )= 2\]

Answer.

\(x(2.2)\approx -1.400\) and \(y(2.2)\approx 1.000\).

\(x(2.4)\approx -1.275\) and \(y(2.4)\approx 2.184\).

\(x(2.6)\approx -0.04759\) and \(y(2.6)\approx 4.846\).

Example 44

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 44)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -t^{2} y^{2} - t x - 3\hspace{2em}x( 1 )= -1\]

\[y'= 4 \, t^{2} x - 4 \, t y + 3\hspace{2em}y( 1 )= 1\]

Answer.

\(x(1.2)\approx -1.600\) and \(y(1.2)\approx 0.0000\).

\(x(1.4)\approx -1.816\) and \(y(1.4)\approx -1.243\).

\(x(1.6)\approx -2.513\) and \(y(1.6)\approx -2.098\).

Example 45

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 45)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -x^{2} y + t y + 2\hspace{2em}x( -1 )= 2\]

\[y'= x^{2} y^{2} - 3 \, t^{2} y + 2\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.90)\approx 1.700\) and \(y(-0.90)\approx 1.300\).

\(x(-0.80)\approx 1.407\) and \(y(-0.80)\approx 1.673\).

\(x(-0.70)\approx 1.142\) and \(y(-0.70)\approx 2.105\).

Example 46

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 46)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -2 \, t^{2} y^{2} - 4 \, x^{2} y + 3\hspace{2em}x( -1 )= -1\]

\[y'= 4 \, t^{2} x^{2} + 2 \, t y^{2} + 2\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.90)\approx -0.5000\) and \(y(-0.90)\approx -0.6000\).

\(x(-0.80)\approx -0.1983\) and \(y(-0.80)\approx -0.3838\).

\(x(-0.70)\approx 0.08886\) and \(y(-0.70)\approx -0.1973\).

Example 47

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 47)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -x^{2} y - t y^{2} - 3\hspace{2em}x( 0 )= 1\]

\[y'= 4 \, x^{2} y^{2} + 2 \, t x - 1\hspace{2em}y( 0 )= -2\]

Answer.

\(x(0.20)\approx 0.8000\) and \(y(0.20)\approx 1.000\).

\(x(0.40)\approx 0.03200\) and \(y(0.40)\approx 1.376\).

\(x(0.60)\approx -0.7197\) and \(y(0.60)\approx 1.183\).

Example 48

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 48)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= 3 \, x^{2} y + 3 \, t y^{2} + 3\hspace{2em}x( 2 )= -1\]

\[y'= t^{2} x^{2} + t^{2} y^{2}\hspace{2em}y( 2 )= -1\]

Answer.

\(x(2.1)\approx -0.4000\) and \(y(2.1)\approx -0.2000\).

\(x(2.2)\approx -0.08440\) and \(y(2.2)\approx -0.1118\).

\(x(2.3)\approx 0.2236\) and \(y(2.3)\approx -0.1023\).

Example 49

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 49)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t x^{2} + 2 \, t^{2} y - 2\hspace{2em}x( 0 )= -1\]

\[y'= -4 \, t^{2} y^{2} + 2 \, x y^{2} - 3\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.10)\approx -1.200\) and \(y(0.10)\approx 0.5000\).

\(x(0.20)\approx -1.341\) and \(y(0.20)\approx 0.1390\).

\(x(0.30)\approx -1.396\) and \(y(0.30)\approx -0.1665\).

Example 50

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 50)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -3 \, t^{2} x^{2} - 2 \, x^{2} y^{2}\hspace{2em}x( 0 )= 1\]

\[y'= -2 \, x^{2} y^{2} + t^{2} y + 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.10)\approx 0.2000\) and \(y(0.10)\approx 1.400\).

\(x(0.20)\approx 0.1842\) and \(y(0.20)\approx 1.586\).

\(x(0.30)\approx 0.1667\) and \(y(0.30)\approx 1.775\).

Example 51

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 51)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -2 \, t x^{2} + x y^{2} - 1\hspace{2em}x( -1 )= 1\]

\[y'= -3 \, t x^{2} - 4 \, x^{2} y - 2\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.90)\approx 1.200\) and \(y(-0.90)\approx -0.5000\).

\(x(-0.80)\approx 1.389\) and \(y(-0.80)\approx -0.02320\).

\(x(-0.70)\approx 1.598\) and \(y(-0.70)\approx 0.2579\).

Example 52

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 52)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= 2 \, t x + 3 \, x y\hspace{2em}x( 2 )= 1\]

\[y'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 1\hspace{2em}y( 2 )= 0\]

Answer.

\(x(2.1)\approx 1.400\) and \(y(2.1)\approx -0.7000\).

\(x(2.2)\approx 1.694\) and \(y(2.2)\approx -1.780\).

\(x(2.3)\approx 1.535\) and \(y(2.3)\approx -2.415\).

Example 53

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 53)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x^{2} - 4 \, t y^{2} + 2\hspace{2em}x( -2 )= 0\]

\[y'= -4 \, t x - 2 \, t y - 2\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.8)\approx 0.4000\) and \(y(-1.8)\approx -0.4000\).

\(x(-1.6)\approx 1.341\) and \(y(-1.6)\approx -0.5120\).

\(x(-1.4)\approx 4.841\) and \(y(-1.4)\approx 0.4774\).

Example 54

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 54)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x^{2} - 4 \, x^{2} y - 3\hspace{2em}x( 0 )= 0\]

\[y'= -t^{2} x^{2} + x y^{2} + 2\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.10)\approx -0.3000\) and \(y(0.10)\approx 0.2000\).

\(x(0.20)\approx -0.6069\) and \(y(0.20)\approx 0.3987\).

\(x(0.30)\approx -0.9613\) and \(y(0.30)\approx 0.5876\).

Example 55

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 55)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -t^{2} y^{2} + 4 \, t x^{2} - 2\hspace{2em}x( 1 )= 0\]

\[y'= -4 \, t x + 4 \, t y - 3\hspace{2em}y( 1 )= 2\]

Answer.

\(x(1.1)\approx -0.6000\) and \(y(1.1)\approx 2.500\).

\(x(1.2)\approx -1.398\) and \(y(1.2)\approx 3.564\).

\(x(1.3)\approx -2.489\) and \(y(1.3)\approx 5.646\).

Example 56

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 56)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, t^{2} y + 2 \, x y - 1\hspace{2em}x( 0 )= 2\]

\[y'= 2 \, t x - 2 \, x y + 1\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.10)\approx 2.300\) and \(y(0.10)\approx 0.7000\).

\(x(0.20)\approx 2.524\) and \(y(0.20)\approx 0.5240\).

\(x(0.30)\approx 2.695\) and \(y(0.30)\approx 0.4604\).

Example 57

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 57)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2\hspace{2em}x( 0 )= 1\]

\[y'= t x + 3 \, t y - 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.10)\approx 0.0000\) and \(y(0.10)\approx 1.800\).

\(x(0.20)\approx 0.2032\) and \(y(0.20)\approx 1.654\).

\(x(0.30)\approx 0.3803\) and \(y(0.30)\approx 1.557\).

Example 58

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 58)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -2 \, x^{2} y - 2 \, t y + 3\hspace{2em}x( -1 )= -1\]

\[y'= -t^{2} x^{2} + 3 \, t y^{2} + 1\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.80)\approx -0.4000\) and \(y(-0.80)\approx -4.400\).

\(x(-0.60)\approx -0.9264\) and \(y(-0.60)\approx -13.51\).

\(x(-0.40)\approx 1.069\) and \(y(-0.40)\approx -79.11\).

Example 59

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 59)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= 4 \, x^{2} y^{2} - t^{2} x - 2\hspace{2em}x( 2 )= 1\]

\[y'= 3 \, t^{2} x^{2} - 3 \, x y^{2} - 3\hspace{2em}y( 2 )= 0\]

Answer.

\(x(2.1)\approx 0.4000\) and \(y(2.1)\approx 0.9000\).

\(x(2.2)\approx 0.07544\) and \(y(2.2)\approx 0.7145\).

\(x(2.3)\approx -0.1599\) and \(y(2.3)\approx 0.4112\).

Example 60

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 60)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, x y^{2} - 2 \, t x + 2\hspace{2em}x( 0 )= 1\]

\[y'= -4 \, t^{2} x^{2} - 3 \, x^{2} y + 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx 1.500\) and \(y(0.10)\approx -0.6000\).

\(x(0.20)\approx 1.832\) and \(y(0.20)\approx -0.1040\).

\(x(0.30)\approx 1.965\) and \(y(0.30)\approx 0.04701\).

Example 61

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 61)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= t^{2} x^{2} + 3 \, x^{2} y + 2\hspace{2em}x( -1 )= 1\]

\[y'= -4 \, t x^{2} + 3 \, t y^{2} - 2\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.80)\approx 1.000\) and \(y(-0.80)\approx -1.200\).

\(x(-0.60)\approx 0.8080\) and \(y(-0.60)\approx -1.651\).

\(x(-0.40)\approx 0.6082\) and \(y(-0.40)\approx -2.719\).

Example 62

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 62)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x - t y - 3\hspace{2em}x( -2 )= 0\]

\[y'= -t^{2} x^{2} - 4 \, x y^{2} + 2\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.8)\approx -0.6000\) and \(y(-1.8)\approx 0.4000\).

\(x(-1.6)\approx -2.222\) and \(y(-1.6)\approx 0.6435\).

\(x(-1.4)\approx -6.030\) and \(y(-1.4)\approx -0.7490\).

Example 63

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 63)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x^{2} + 3 \, t y^{2} - 2\hspace{2em}x( -1 )= 0\]

\[y'= 2 \, x^{2} y - t y^{2} + 3\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.80)\approx -1.000\) and \(y(-0.80)\approx -0.2000\).

\(x(-0.60)\approx -1.163\) and \(y(-0.60)\approx 0.3264\).

\(x(-0.40)\approx -1.407\) and \(y(-0.40)\approx 1.116\).

Example 64

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 64)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= t^{2} y^{2} + 2 \, x^{2} y^{2} + 1\hspace{2em}x( 0 )= -1\]

\[y'= 3 \, t^{2} y^{2} + t x + 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.20)\approx -0.8000\) and \(y(0.20)\approx 0.2000\).

\(x(0.40)\approx -0.5894\) and \(y(0.40)\approx 0.3690\).

\(x(0.60)\approx -0.3662\) and \(y(0.60)\approx 0.5349\).

Example 65

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 65)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -3 \, t y^{2} - 4 \, t x + 3\hspace{2em}x( -1 )= -2\]

\[y'= -3 \, x^{2} y^{2} + 2 \, t y - 3\hspace{2em}y( -1 )= 2\]

Answer.

\(x(-0.90)\approx -1.300\) and \(y(-0.90)\approx -3.500\).

\(x(-0.80)\approx 1.839\) and \(y(-0.80)\approx -9.381\).

\(x(-0.70)\approx 23.85\) and \(y(-0.70)\approx -97.51\).

Example 66

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 66)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= x y^{2} - 4 \, t x - 2\hspace{2em}x( 2 )= 1\]

\[y'= -4 \, t^{2} x^{2} + 2 \, t^{2} y^{2} - 2\hspace{2em}y( 2 )= -1\]

Answer.

\(x(2.1)\approx 0.1000\) and \(y(2.1)\approx -2.000\).

\(x(2.2)\approx -0.1440\) and \(y(2.2)\approx 1.310\).

\(x(2.3)\approx -0.2420\) and \(y(2.3)\approx 2.732\).

Example 67

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 67)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x - 4 \, t y^{2} - 3\hspace{2em}x( 2 )= 0\]

\[y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y - 2\hspace{2em}y( 2 )= 1\]

Answer.

\(x(2.1)\approx -1.100\) and \(y(2.1)\approx -0.4000\).

\(x(2.2)\approx -3.475\) and \(y(2.2)\approx -1.138\).

\(x(2.3)\approx -11.64\) and \(y(2.3)\approx -11.37\).

Example 68

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 68)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= t^{2} x^{2} + x y^{2} + 2\hspace{2em}x( -1 )= -2\]

\[y'= -2 \, t y^{2} + 3 \, x y + 1\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.80)\approx -1.200\) and \(y(-0.80)\approx 0.4000\).

\(x(-0.60)\approx -0.6541\) and \(y(-0.60)\approx 0.3632\).

\(x(-0.40)\approx -0.2405\) and \(y(-0.40)\approx 0.4523\).

Example 69

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 69)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x - 3 \, x^{2} y + 2\hspace{2em}x( -1 )= -1\]

\[y'= 2 \, t^{2} x^{2} - 4 \, x^{2} y - 3\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.80)\approx -1.600\) and \(y(-0.80)\approx 0.0000\).

\(x(-0.60)\approx -1.610\) and \(y(-0.60)\approx 0.05536\).

\(x(-0.40)\approx -1.527\) and \(y(-0.40)\approx -0.2863\).

Example 70

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 70)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -2 \, t^{2} x^{2} - x y^{2} + 3\hspace{2em}x( -2 )= -1\]

\[y'= 3 \, t x^{2} - x^{2} y - 3\hspace{2em}y( -2 )= 1\]

Answer.

\(x(-1.9)\approx -1.400\) and \(y(-1.9)\approx 0.0000\).

\(x(-1.8)\approx -2.515\) and \(y(-1.8)\approx -1.417\).

\(x(-1.7)\approx -5.809\) and \(y(-1.7)\approx -4.237\).

Example 71

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 71)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -2 \, t^{2} x^{2} + 4 \, t y + 2\hspace{2em}x( 0 )= 1\]

\[y'= -4 \, t^{2} y^{2} + 3 \, t^{2} x - 2\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.10)\approx 1.200\) and \(y(0.10)\approx -0.2000\).

\(x(0.20)\approx 1.389\) and \(y(0.20)\approx -0.3966\).

\(x(0.30)\approx 1.542\) and \(y(0.30)\approx -0.5824\).

Example 72

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 72)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y^{2} - 2 \, x y - 2\hspace{2em}x( 0 )= 1\]

\[y'= -4 \, x y^{2} - t y + 3\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.20)\approx 0.2000\) and \(y(0.20)\approx 0.8000\).

\(x(0.40)\approx -0.2794\) and \(y(0.40)\approx 1.266\).

\(x(0.60)\approx -0.6917\) and \(y(0.60)\approx 2.122\).

Example 73

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 73)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= 3 \, t^{2} x + 2 \, x y + 1\hspace{2em}x( 0 )= 1\]

\[y'= 2 \, t^{2} y^{2} - 4 \, x^{2} y + 3\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.20)\approx 0.8000\) and \(y(0.20)\approx 0.4000\).

\(x(0.40)\approx 1.147\) and \(y(0.40)\approx 0.7978\).

\(x(0.60)\approx 1.823\) and \(y(0.60)\approx 0.5986\).

Example 74

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 74)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= t x^{2} - 3 \, t y^{2} - 2\hspace{2em}x( 1 )= -1\]

\[y'= -2 \, t x + 2 \, t y - 3\hspace{2em}y( 1 )= -1\]

Answer.

\(x(1.2)\approx -1.800\) and \(y(1.2)\approx -1.600\).

\(x(1.4)\approx -3.266\) and \(y(1.4)\approx -2.104\).

\(x(1.6)\approx -4.398\) and \(y(1.6)\approx -2.053\).

Example 75

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 75)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 3 \, t^{2} y^{2} + x y^{2} + 3\hspace{2em}x( 1 )= 2\]

\[y'= 2 \, t^{2} y^{2} + t^{2} x - 1\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.1)\approx 2.300\) and \(y(1.1)\approx 0.1000\).

\(x(1.2)\approx 2.606\) and \(y(1.2)\approx 0.2807\).

\(x(1.3)\approx 2.961\) and \(y(1.3)\approx 0.5787\).

Example 76

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 76)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= 4 \, t^{2} y^{2} + 4 \, t^{2} x + 3\hspace{2em}x( 1 )= 1\]

\[y'= 3 \, t x^{2} - 4 \, x y - 2\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.2)\approx 2.400\) and \(y(1.2)\approx 0.2000\).

\(x(1.4)\approx 5.811\) and \(y(1.4)\approx 3.563\).

\(x(1.6)\approx 35.43\) and \(y(1.6)\approx 14.96\).

Example 77

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 77)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -2 \, t x^{2} + 2 \, t y - 1\hspace{2em}x( 1 )= 2\]

\[y'= 3 \, t^{2} y^{2} + 3 \, t^{2} x - 2\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.2)\approx -0.6000\) and \(y(1.2)\approx 1.200\).

\(x(1.4)\approx -0.3968\) and \(y(1.4)\approx 1.526\).

\(x(1.6)\approx 0.1695\) and \(y(1.6)\approx 3.397\).

Example 78

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 78)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= t^{2} x^{2} - 3 \, t^{2} y^{2} - 2\hspace{2em}x( -2 )= -2\]

\[y'= t^{2} x - x^{2} y\hspace{2em}y( -2 )= 1\]

Answer.

\(x(-1.9)\approx -1.800\) and \(y(-1.9)\approx -0.2000\).

\(x(-1.8)\approx -0.8737\) and \(y(-1.8)\approx -0.7850\).

\(x(-1.7)\approx -1.425\) and \(y(-1.7)\approx -1.008\).

Example 79

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 79)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -2 \, t^{2} y - 3 \, x y\hspace{2em}x( -1 )= 1\]

\[y'= 4 \, t^{2} y^{2} - 3 \, x^{2} y + 3\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.80)\approx 3.000\) and \(y(-0.80)\approx 3.000\).

\(x(-0.60)\approx -3.168\) and \(y(-0.60)\approx -7.992\).

\(x(-0.40)\approx -17.21\) and \(y(-0.40)\approx 59.13\).

Example 80

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 80)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t^{2} y^{2} + 4 \, t x^{2} + 1\hspace{2em}x( 0 )= 2\]

\[y'= 4 \, t x + 2 \, t y\hspace{2em}y( 0 )= -2\]

Answer.

\(x(0.10)\approx 2.100\) and \(y(0.10)\approx -2.000\).

\(x(0.20)\approx 2.392\) and \(y(0.20)\approx -1.956\).

\(x(0.30)\approx 3.012\) and \(y(0.30)\approx -1.843\).

Example 81

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 81)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -x^{2} y - 3 \, t y^{2} + 3\hspace{2em}x( -1 )= 0\]

\[y'= -3 \, t^{2} y + 2 \, x y^{2}\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.80)\approx 1.200\) and \(y(-0.80)\approx -0.4000\).

\(x(-0.60)\approx 1.992\) and \(y(-0.60)\approx -0.1696\).

\(x(-0.40)\approx 2.737\) and \(y(-0.40)\approx -0.1100\).

Example 82

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 82)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, x^{2} y^{2} - t y + 3\hspace{2em}x( 0 )= 2\]

\[y'= t^{2} x - x y^{2} + 3\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.10)\approx 3.500\) and \(y(0.10)\approx 1.100\).

\(x(0.20)\approx 8.236\) and \(y(0.20)\approx 0.9800\).

\(x(0.30)\approx 28.06\) and \(y(0.30)\approx 0.5220\).

Example 83

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 83)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -x^{2} y - 2 \, t y + 1\hspace{2em}x( 1 )= 0\]

\[y'= -2 \, t^{2} x + 4 \, t y^{2} - 1\hspace{2em}y( 1 )= -1\]

Answer.

\(x(1.1)\approx 0.3000\) and \(y(1.1)\approx -0.7000\).

\(x(1.2)\approx 0.5603\) and \(y(1.2)\approx -0.6570\).

\(x(1.3)\approx 0.8386\) and \(y(1.3)\approx -0.7112\).

Example 84

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 84)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y^{2} - 3 \, x y^{2} + 2\hspace{2em}x( 2 )= -2\]

\[y'= -2 \, t y^{2} - 2 \, x y^{2} - 2\hspace{2em}y( 2 )= 1\]

Answer.

\(x(2.1)\approx -2.400\) and \(y(2.1)\approx 0.8000\).

\(x(2.2)\approx -2.586\) and \(y(2.2)\approx 0.6384\).

\(x(2.3)\approx -2.662\) and \(y(2.3)\approx 0.4699\).

Example 85

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 85)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -3 \, t y^{2} - 2 \, t x - 2\hspace{2em}x( 2 )= -2\]

\[y'= 4 \, t^{2} y^{2} - 4 \, t x^{2} + 1\hspace{2em}y( 2 )= 1\]

Answer.

\(x(2.1)\approx -2.000\) and \(y(2.1)\approx -0.5000\).

\(x(2.2)\approx -1.518\) and \(y(2.2)\approx -3.319\).

\(x(2.3)\approx -8.320\) and \(y(2.3)\approx 16.08\).

Example 86

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 86)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y^{2} - 2 \, t^{2} x + 1\hspace{2em}x( 0 )= 2\]

\[y'= -4 \, t y^{2} + 4 \, x y^{2} - 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx 2.100\) and \(y(0.10)\approx -0.3000\).

\(x(0.20)\approx 2.196\) and \(y(0.20)\approx -0.3280\).

\(x(0.30)\approx 2.279\) and \(y(0.30)\approx -0.3421\).

Example 87

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 87)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} + 2 \, t^{2} x + 1\hspace{2em}x( -1 )= 1\]

\[y'= 4 \, t^{2} y^{2} + 4 \, t^{2} x - 3\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.80)\approx 0.8000\) and \(y(-0.80)\approx 0.0000\).

\(x(-0.60)\approx 1.205\) and \(y(-0.60)\approx -0.1904\).

\(x(-0.40)\approx 1.536\) and \(y(-0.40)\approx -0.4330\).

Example 88

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 88)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -4 \, t^{2} x^{2} + 2 \, t y^{2} - 3\hspace{2em}x( 2 )= 0\]

\[y'= -4 \, t^{2} y + 4 \, x y^{2} + 1\hspace{2em}y( 2 )= 2\]

Answer.

\(x(2.1)\approx 1.300\) and \(y(2.1)\approx -1.100\).

\(x(2.2)\approx -1.473\) and \(y(2.2)\approx 1.570\).

\(x(2.3)\approx -4.889\) and \(y(2.3)\approx -2.821\).

Example 89

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 89)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -4 \, t^{2} x^{2} + x y^{2} - 3\hspace{2em}x( -1 )= 1\]

\[y'= -4 \, t^{2} y - x y^{2} - 3\hspace{2em}y( -1 )= 2\]

Answer.

\(x(-0.80)\approx 0.4000\) and \(y(-0.80)\approx -1.000\).

\(x(-0.60)\approx -0.2019\) and \(y(-0.60)\approx -1.168\).

\(x(-0.40)\approx -0.8688\) and \(y(-0.40)\approx -1.377\).

Example 90

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 90)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -2 \, t^{2} x^{2} - 3 \, t^{2} y - 3\hspace{2em}x( 1 )= 1\]

\[y'= 4 \, t^{2} x^{2} + 2 \, t y\hspace{2em}y( 1 )= -1\]

Answer.

\(x(1.1)\approx 0.8000\) and \(y(1.1)\approx -0.8000\).

\(x(1.2)\approx 0.6355\) and \(y(1.2)\approx -0.6662\).

\(x(1.3)\approx 0.5070\) and \(y(1.3)\approx -0.5935\).

Example 91

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 91)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y^{2} + 3 \, t x - 2\hspace{2em}x( 2 )= -2\]

\[y'= 3 \, t y^{2} - 2 \, x y + 2\hspace{2em}y( 2 )= 0\]

Answer.

\(x(2.2)\approx -4.800\) and \(y(2.2)\approx 0.4000\).

\(x(2.4)\approx -12.00\) and \(y(2.4)\approx 1.779\).

\(x(2.6)\approx -40.62\) and \(y(2.6)\approx 15.28\).

Example 92

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 92)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y - 4 \, x^{2} y - 1\hspace{2em}x( 2 )= 0\]

\[y'= -3 \, t^{2} x - 3 \, t y - 1\hspace{2em}y( 2 )= 1\]

Answer.

\(x(2.2)\approx 1.400\) and \(y(2.2)\approx -0.4000\).

\(x(2.4)\approx 1.053\) and \(y(2.4)\approx -4.138\).

\(x(2.6)\approx -5.011\) and \(y(2.6)\approx -2.018\).

Example 93

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 93)

Use Euler's method with \(h=0.10\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y + x^{2} y + 3\hspace{2em}x( 1 )= -2\]

\[y'= -3 \, x y^{2} - 4 \, t y - 1\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.1)\approx -1.700\) and \(y(1.1)\approx -0.1000\).

\(x(1.2)\approx -1.393\) and \(y(1.2)\approx -0.1509\).

\(x(1.3)\approx -1.057\) and \(y(1.3)\approx -0.1690\).

Example 94

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 94)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, t y^{2} - 3 \, t x - 1\hspace{2em}x( 0 )= -1\]

\[y'= -x^{2} y^{2} - 2 \, t x + 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.10)\approx -1.100\) and \(y(0.10)\approx 1.800\).

\(x(0.20)\approx -1.070\) and \(y(0.20)\approx 1.630\).

\(x(0.30)\approx -0.9462\) and \(y(0.30)\approx 1.569\).

Example 95

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 95)

Use Euler's method with \(h=0.10\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -4 \, t^{2} y^{2} + 2 \, t x^{2} - 3\hspace{2em}x( 0 )= 2\]

\[y'= 3 \, t^{2} y - 4 \, x^{2} y + 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.10)\approx 1.700\) and \(y(0.10)\approx 0.7000\).

\(x(0.20)\approx 1.456\) and \(y(0.20)\approx -0.007100\).

\(x(0.30)\approx 1.241\) and \(y(0.30)\approx 0.09883\).

Example 96

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 96)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= 3 \, t x^{2} + 2 \, t y\hspace{2em}x( -2 )= 2\]

\[y'= 3 \, t x^{2} - 4 \, x^{2} y - 3\hspace{2em}y( -2 )= 2\]

Answer.

\(x(-1.9)\approx -1.200\) and \(y(-1.9)\approx -3.900\).

\(x(-1.8)\approx -0.5388\) and \(y(-1.8)\approx -2.774\).

\(x(-1.7)\approx 0.3032\) and \(y(-1.7)\approx -2.909\).

Example 97

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 97)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 3 \, x^{2} y^{2} + 4 \, t y^{2}\hspace{2em}x( -1 )= 2\]

\[y'= t^{2} y^{2} + 3 \, t x - 1\hspace{2em}y( -1 )= 2\]

Answer.

\(x(-0.90)\approx 5.200\) and \(y(-0.90)\approx 1.700\).

\(x(-0.80)\approx 27.60\) and \(y(-0.80)\approx 0.4301\).

\(x(-0.70)\approx 69.83\) and \(y(-0.70)\approx -6.283\).

Example 98

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 98)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -3 \, x^{2} y^{2} - t^{2} x + 2\hspace{2em}x( 0 )= 0\]

\[y'= -3 \, t^{2} y^{2} - x^{2} y^{2}\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.20)\approx 0.4000\) and \(y(0.20)\approx -1.000\).

\(x(0.40)\approx 0.7008\) and \(y(0.40)\approx -1.056\).

\(x(0.60)\approx 0.7498\) and \(y(0.60)\approx -1.273\).

Example 99

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 99)

Use Euler's method with \(h=0.10\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -t x + 2 \, t y - 3\hspace{2em}x( 2 )= -2\]

\[y'= t^{2} x^{2} + 2 \, x y^{2}\hspace{2em}y( 2 )= 0\]

Answer.

\(x(2.1)\approx -1.900\) and \(y(2.1)\approx 1.600\).

\(x(2.2)\approx -1.129\) and \(y(2.2)\approx 2.219\).

\(x(2.3)\approx -0.2042\) and \(y(2.3)\approx 1.724\).

Example 100

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 100)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -t^{2} y - x^{2} y + 3\hspace{2em}x( -1 )= 0\]

\[y'= 2 \, x^{2} y + t y^{2} + 3\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.80)\approx 1.000\) and \(y(-0.80)\approx -2.200\).

\(x(-0.60)\approx 2.322\) and \(y(-0.60)\approx -3.254\).

\(x(-0.40)\approx 6.664\) and \(y(-0.40)\approx -10.94\).