N3m - Programming Euler’s method. Implement Euler’s method using technology.

Example 1

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 1)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-1.4)\approx -1.357\) and \(y(-1.4)\approx -0.1167\).

Example 2

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 2)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx -0.6627\) and \(y(-0.40)\approx -1.231\).

Example 3

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 3)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

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

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

Answer.

\(x(2.3)\approx 1.283\) and \(y(2.3)\approx -1.727\).

Example 4

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 4)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx 1.105\) and \(y(0.60)\approx -0.5699\).

Example 5

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 5)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx 0.2463\) and \(y(0.60)\approx 1.277\).

Example 6

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 6)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

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

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

Answer.

\(x(1.3)\approx 8.490\) and \(y(1.3)\approx 4.582\).

Example 7

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 7)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -1.154\) and \(y(0.30)\approx -1.667\).

Example 8

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 8)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -1.827\) and \(y(1.3)\approx -3.460\).

Example 9

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 9)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 0.1268\) and \(y(0.30)\approx 3.912\).

Example 10

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 10)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx -1.751\) and \(y(0.60)\approx 0.3600\).

Example 11

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 11)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

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

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

Answer.

\(x(2.3)\approx -0.7502\) and \(y(2.3)\approx 0.2308\).

Example 12

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 12)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -0.6836\) and \(y(1.3)\approx 0.2074\).

Example 13

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 13)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx -1.367\) and \(y(-1.7)\approx -1.263\).

Example 14

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 14)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 5.716\) and \(y(-0.70)\approx 0.08035\).

Example 15

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 15)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 1.110\) and \(y(-0.70)\approx -6.628\).

Example 16

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 16)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 1.620\) and \(y(-0.70)\approx 0.7901\).

Example 17

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 17)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx -1.968\) and \(y(-0.70)\approx 0.4260\).

Example 18

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 18)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx 1.978\) and \(y(-1.7)\approx -0.8414\).

Example 19

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 19)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 0.2759\) and \(y(0.30)\approx -1.106\).

Example 20

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 20)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -1.555\) and \(y(0.30)\approx 1.088\).

Example 21

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 21)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

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

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

Answer.

\(x(2.6)\approx 0.2049\) and \(y(2.6)\approx -3.063\).

Example 22

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 22)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx 1.590\) and \(y(0.60)\approx 2.007\).

Example 23

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 23)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx 13.13\) and \(y(1.3)\approx -5.796\).

Example 24

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 24)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -0.2153\) and \(y(1.3)\approx -1.094\).

Example 25

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 25)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 4.729\) and \(y(-0.70)\approx -4.399\).

Example 26

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 26)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.4)\approx -0.8088\) and \(y(-1.4)\approx 0.6344\).

Example 27

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 27)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -10.36\) and \(y(1.3)\approx -13.47\).

Example 28

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 28)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx 2.456\) and \(y(-1.7)\approx -0.1348\).

Example 29

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 29)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 0.04812\) and \(y(0.30)\approx 3.033\).

Example 30

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 30)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 1.044\) and \(y(-0.70)\approx -7.591\).

Example 31

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 31)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 0.7309\) and \(y(-0.70)\approx -0.7151\).

Example 32

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 32)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

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

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

Answer.

\(x(0.60)\approx -1.475\) and \(y(0.60)\approx 0.7611\).

Example 33

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 33)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

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

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

Answer.

\(x(0.60)\approx 2.949\) and \(y(0.60)\approx 0.03386\).

Example 34

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 34)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

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

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

Answer.

\(x(1.3)\approx 4.159\) and \(y(1.3)\approx -6.837\).

Example 35

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 35)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 0.2896\) and \(y(-0.40)\approx 0.8183\).

Example 36

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 36)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 0.4277\) and \(y(-0.40)\approx 1.001\).

Example 37

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 37)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx 24.13\) and \(y(2.3)\approx 19.08\).

Example 38

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 38)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(1.6)\approx -12.89\) and \(y(1.6)\approx -3.994\).

Example 39

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 39)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx -4.541\) and \(y(0.60)\approx 3.043\).

Example 40

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 40)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx 0.3002\) and \(y(-1.7)\approx 0.08715\).

Example 41

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 41)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

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

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

Answer.

\(x(1.3)\approx -0.2426\) and \(y(1.3)\approx 0.5265\).

Example 42

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 42)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -0.8581\) and \(y(0.30)\approx -1.293\).

Example 43

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 43)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

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

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

Answer.

\(x(1.6)\approx -0.2420\) and \(y(1.6)\approx -1.360\).

Example 44

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 44)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(1.6)\approx -2.804\) and \(y(1.6)\approx -2.313\).

Example 45

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 45)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 1.127\) and \(y(-0.70)\approx 2.264\).

Example 46

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 46)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx -0.01399\) and \(y(-0.70)\approx -0.3053\).

Example 47

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 47)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx -0.7424\) and \(y(0.60)\approx -1.150\).

Example 48

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 48)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx 0.2921\) and \(y(2.3)\approx -0.3078\).

Example 49

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 49)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -1.318\) and \(y(0.30)\approx -0.08504\).

Example 50

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 50)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 0.3189\) and \(y(0.30)\approx 1.939\).

Example 51

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 51)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx 1.617\) and \(y(-0.70)\approx 0.1672\).

Example 52

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 52)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx 1.303\) and \(y(2.3)\approx -2.044\).

Example 53

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 53)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

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

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

Answer.

\(x(-0.40)\approx 0.4149\) and \(y(-0.40)\approx -0.04983\).

Example 54

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 54)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -1.041\) and \(y(0.30)\approx 0.5719\).

Example 55

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 55)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -4.322\) and \(y(1.3)\approx 8.902\).

Example 56

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 56)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 2.697\) and \(y(0.30)\approx 0.5338\).

Example 57

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 57)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

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

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

Answer.

\(x(1.6)\approx 0.7354\) and \(y(1.6)\approx 1.314\).

Example 58

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 58)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.4)\approx 0.04515\) and \(y(-1.4)\approx -0.1416\).

Example 59

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 59)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx -0.06677\) and \(y(2.3)\approx -0.01825\).

Example 60

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 60)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 1.774\) and \(y(0.30)\approx -0.06923\).

Example 61

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 61)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 0.2728\) and \(y(-0.40)\approx -18.52\).

Example 62

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 62)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

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

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

Answer.

\(x(2.3)\approx -0.4207\) and \(y(2.3)\approx 0.07870\).

Example 63

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 63)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx -1.342\) and \(y(-0.40)\approx 1.229\).

Example 64

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 64)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx -0.3568\) and \(y(0.60)\approx 0.5305\).

Example 65

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 65)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-0.70)\approx -3.216\) and \(y(-0.70)\approx -0.4657\).

Example 66

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 66)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx -0.1338\) and \(y(2.3)\approx -0.5210\).

Example 67

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 67)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.7)\approx 2.631\) and \(y(-1.7)\approx 15.42\).

Example 68

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 68)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx -0.4884\) and \(y(-0.40)\approx 0.5670\).

Example 69

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 69)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx -1.223\) and \(y(-0.40)\approx -0.2217\).

Example 70

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 70)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

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

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

Answer.

\(x(2.3)\approx 2.550\) and \(y(2.3)\approx -3.703\).

Example 71

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 71)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 1.500\) and \(y(0.30)\approx -0.5708\).

Example 72

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 72)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx -0.9897\) and \(y(0.60)\approx 4.249\).

Example 73

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 73)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx 2.032\) and \(y(0.60)\approx 0.2495\).

Example 74

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 74)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(1.6)\approx -2.974\) and \(y(1.6)\approx -1.480\).

Example 75

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 75)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx 3.179\) and \(y(1.3)\approx 0.8140\).

Example 76

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 76)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

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

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

Answer.

\(x(-0.40)\approx 0.2873\) and \(y(-0.40)\approx 3.760\).

Example 77

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 77)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(1.6)\approx -0.4124\) and \(y(1.6)\approx -0.6430\).

Example 78

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 78)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx -1.110\) and \(y(-1.7)\approx -0.5731\).

Example 79

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 79)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 1.210\) and \(y(-0.40)\approx 0.4858\).

Example 80

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 80)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 3.780\) and \(y(0.30)\approx -1.683\).

Example 81

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 81)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 2.655\) and \(y(-0.40)\approx -0.2088\).

Example 82

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 82)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.4)\approx -0.1238\) and \(y(-1.4)\approx -2.511\).

Example 83

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 83)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx 0.9034\) and \(y(1.3)\approx -0.7827\).

Example 84

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 84)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

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

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

Answer.

\(x(2.3)\approx 0.6802\) and \(y(2.3)\approx -0.3454\).

Example 85

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 85)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx -2.528\) and \(y(2.3)\approx -1.587\).

Example 86

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 86)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 2.266\) and \(y(0.30)\approx -0.4167\).

Example 87

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 87)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx 1.061\) and \(y(-0.40)\approx -0.8239\).

Example 88

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 88)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx -0.6301\) and \(y(2.3)\approx 0.05394\).

Example 89

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 89)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(-0.40)\approx -1.046\) and \(y(-0.40)\approx -0.7226\).

Example 90

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 90)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx 0.5660\) and \(y(1.3)\approx -0.6374\).

Example 91

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 91)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.7)\approx -0.1948\) and \(y(-1.7)\approx -5.823\).

Example 92

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 92)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(2.6)\approx 0.1872\) and \(y(2.6)\approx -0.03408\).

Example 93

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 93)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(1.3)\approx -1.039\) and \(y(1.3)\approx -0.1512\).

Example 94

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 94)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx -0.7752\) and \(y(0.30)\approx 1.739\).

Example 95

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 95)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(0.30)\approx 1.302\) and \(y(0.30)\approx 0.07809\).

Example 96

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 96)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(-1.7)\approx 0.6312\) and \(y(-1.7)\approx -0.8663\).

Example 97

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 97)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

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

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

Answer.

\(x(-0.40)\approx 1.268\) and \(y(-0.40)\approx -0.3093\).

Example 98

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 98)

Use technology to implement Euler's method with \(h=0.020\) 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.

\(x(0.60)\approx 0.5995\) and \(y(0.60)\approx -1.496\).

Example 99

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 99)

Use technology to implement Euler's method with \(h=0.010\) 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.

\(x(2.3)\approx -0.4594\) and \(y(2.3)\approx 1.503\).

Example 100

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 100)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

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

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

Answer.

\(x(1.3)\approx -0.04375\) and \(y(1.3)\approx -1.020\).