D1 - Laplace transform. Compute the Laplace transform of a function from the definition.

Example 1

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 1)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, t^{3} \right\}= \frac{18}{s^{4}}\]

Example 2

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 2)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, e^{\left(3 \, t\right)} - 2 \, u\left(t - 1\right) \right\}= -\frac{2 \, e^{\left(-s\right)}}{s} + \frac{2}{s - 3}\]

Example 3

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 3)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \delta\left(t - 3\right) - 4 \, e^{\left(3 \, t\right)} \right\}= -\frac{4}{s - 3} + 2 \, e^{\left(-3 \, s\right)}\]

Example 4

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 4)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \cos\left(t\right) \right\}= \frac{2 \, s}{s^{2} + 1}\]

Example 5

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 5)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \cos\left(t\right) \right\}= -\frac{3 \, s}{s^{2} + 1}\]

Example 6

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 6)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \delta\left(t - 3\right) + 4 \, e^{\left(2 \, t\right)} \right\}= \frac{4}{s - 2} - 2 \, e^{\left(-3 \, s\right)}\]

Example 7

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 7)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \delta\left(t - 3\right) + 4 \, e^{\left(2 \, t\right)} \right\}= \frac{4}{s - 2} + 4 \, e^{\left(-3 \, s\right)}\]

Example 8

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 8)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \cos\left(3 \, t\right) \right\}= \frac{4 \, s}{s^{2} + 9}\]

Example 9

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 9)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \cos\left(2 \, t\right) \right\}= \frac{2 \, s}{s^{2} + 4}\]

Example 10

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 10)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \sin\left(t\right) \right\}= \frac{2}{s^{2} + 1}\]

Example 11

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 11)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \sin\left(t\right) \right\}= -\frac{2}{s^{2} + 1}\]

Example 12

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 12)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \sin\left(3 \, t\right) \right\}= -\frac{9}{s^{2} + 9}\]

Example 13

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 13)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, t^{2} \right\}= -\frac{8}{s^{3}}\]

Example 14

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 14)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -\sin\left(2 \, t\right) \right\}= -\frac{2}{s^{2} + 4}\]

Example 15

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 15)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \delta\left(t - 2\right) + 3 \, e^{t} \right\}= \frac{3}{s - 1} - 3 \, e^{\left(-2 \, s\right)}\]

Example 16

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 16)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, t^{3} \right\}= -\frac{18}{s^{4}}\]

Example 17

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 17)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \sin\left(3 \, t\right) \right\}= -\frac{6}{s^{2} + 9}\]

Example 18

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 18)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \delta\left(t - 3\right) + 2 \, e^{\left(2 \, t\right)} \right\}= \frac{2}{s - 2} + 2 \, e^{\left(-3 \, s\right)}\]

Example 19

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 19)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \delta\left(t - 1\right) - 4 \, e^{t} \right\}= -\frac{4}{s - 1} + 3 \, e^{\left(-s\right)}\]

Example 20

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 20)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \sin\left(t\right) \right\}= -\frac{3}{s^{2} + 1}\]

Example 21

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 21)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \delta\left(t - 3\right) - e^{t} \right\}= -\frac{1}{s - 1} - 3 \, e^{\left(-3 \, s\right)}\]

Example 22

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 22)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \cos\left(3 \, t\right) \right\}= -\frac{2 \, s}{s^{2} + 9}\]

Example 23

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 23)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -\cos\left(t\right) \right\}= -\frac{s}{s^{2} + 1}\]

Example 24

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 24)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \sin\left(2 \, t\right) \right\}= -\frac{8}{s^{2} + 4}\]

Example 25

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 25)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \sin\left(3 \, t\right) \right\}= -\frac{12}{s^{2} + 9}\]

Example 26

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 26)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, e^{\left(3 \, t\right)} - 3 \, u\left(t - 2\right) \right\}= -\frac{3 \, e^{\left(-2 \, s\right)}}{s} + \frac{3}{s - 3}\]

Example 27

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 27)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, t^{2} \right\}= -\frac{8}{s^{3}}\]

Example 28

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 28)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \delta\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} \right\}= -\frac{4}{s - 3} + 2 \, e^{\left(-s\right)}\]

Example 29

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 29)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \sin\left(2 \, t\right) \right\}= \frac{6}{s^{2} + 4}\]

Example 30

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 30)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \delta\left(t - 3\right) - 4 \, e^{t} \right\}= -\frac{4}{s - 1} - 3 \, e^{\left(-3 \, s\right)}\]

Example 31

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 31)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \cos\left(2 \, t\right) \right\}= -\frac{3 \, s}{s^{2} + 4}\]

Example 32

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 32)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \delta\left(t - 1\right) + 3 \, e^{\left(3 \, t\right)} \right\}= \frac{3}{s - 3} + 4 \, e^{\left(-s\right)}\]

Example 33

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 33)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \delta\left(t - 1\right) - 2 \, e^{\left(3 \, t\right)} \right\}= -\frac{2}{s - 3} + 4 \, e^{\left(-s\right)}\]

Example 34

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 34)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \cos\left(3 \, t\right) \right\}= \frac{4 \, s}{s^{2} + 9}\]

Example 35

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 35)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \delta\left(t - 3\right) - 4 \, e^{\left(3 \, t\right)} \right\}= -\frac{4}{s - 3} + 4 \, e^{\left(-3 \, s\right)}\]

Example 36

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 36)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \cos\left(t\right) \right\}= \frac{2 \, s}{s^{2} + 1}\]

Example 37

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 37)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, t^{2} \right\}= \frac{6}{s^{3}}\]

Example 38

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 38)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -t^{3} \right\}= -\frac{6}{s^{4}}\]

Example 39

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 39)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ e^{t} + 2 \, u\left(t - 1\right) \right\}= \frac{2 \, e^{\left(-s\right)}}{s} + \frac{1}{s - 1}\]

Example 40

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 40)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, t^{2} \right\}= -\frac{4}{s^{3}}\]

Example 41

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 41)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \delta\left(t - 2\right) - 4 \, e^{\left(3 \, t\right)} \right\}= -\frac{4}{s - 3} + e^{\left(-2 \, s\right)}\]

Example 42

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 42)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, t^{2} \right\}= -\frac{8}{s^{3}}\]

Example 43

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 43)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, e^{\left(3 \, t\right)} - 3 \, u\left(t - 2\right) \right\}= -\frac{3 \, e^{\left(-2 \, s\right)}}{s} - \frac{3}{s - 3}\]

Example 44

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 44)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -\delta\left(t - 2\right) - e^{\left(2 \, t\right)} \right\}= -\frac{1}{s - 2} - e^{\left(-2 \, s\right)}\]

Example 45

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 45)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \sin\left(t\right) \right\}= \frac{3}{s^{2} + 1}\]

Example 46

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 46)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, t \right\}= -\frac{4}{s^{2}}\]

Example 47

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 47)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, t \right\}= \frac{2}{s^{2}}\]

Example 48

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 48)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, t^{2} \right\}= -\frac{6}{s^{3}}\]

Example 49

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 49)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \delta\left(t - 1\right) - e^{\left(3 \, t\right)} \right\}= -\frac{1}{s - 3} - 2 \, e^{\left(-s\right)}\]

Example 50

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 50)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \sin\left(t\right) \right\}= \frac{1}{s^{2} + 1}\]

Example 51

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 51)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, t^{3} \right\}= -\frac{12}{s^{4}}\]

Example 52

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 52)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \cos\left(2 \, t\right) \right\}= -\frac{4 \, s}{s^{2} + 4}\]

Example 53

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 53)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \delta\left(t - 2\right) + 3 \, e^{t} \right\}= \frac{3}{s - 1} + e^{\left(-2 \, s\right)}\]

Example 54

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 54)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \cos\left(t\right) \right\}= \frac{3 \, s}{s^{2} + 1}\]

Example 55

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 55)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, e^{\left(2 \, t\right)} - 2 \, u\left(t - 2\right) \right\}= -\frac{2 \, e^{\left(-2 \, s\right)}}{s} + \frac{4}{s - 2}\]

Example 56

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 56)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ t^{3} \right\}= \frac{6}{s^{4}}\]

Example 57

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 57)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \cos\left(3 \, t\right) \right\}= \frac{s}{s^{2} + 9}\]

Example 58

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 58)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, t \right\}= \frac{2}{s^{2}}\]

Example 59

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 59)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -t \right\}= -\frac{1}{s^{2}}\]

Example 60

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 60)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \cos\left(3 \, t\right) \right\}= -\frac{2 \, s}{s^{2} + 9}\]

Example 61

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 61)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \sin\left(2 \, t\right) \right\}= -\frac{8}{s^{2} + 4}\]

Example 62

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 62)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \sin\left(t\right) \right\}= -\frac{2}{s^{2} + 1}\]

Example 63

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 63)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ t \right\}= \frac{1}{s^{2}}\]

Example 64

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 64)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, t^{3} \right\}= \frac{18}{s^{4}}\]

Example 65

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 65)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \delta\left(t - 3\right) + 2 \, e^{\left(3 \, t\right)} \right\}= \frac{2}{s - 3} + 2 \, e^{\left(-3 \, s\right)}\]

Example 66

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 66)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -\cos\left(2 \, t\right) \right\}= -\frac{s}{s^{2} + 4}\]

Example 67

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 67)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \delta\left(t - 3\right) + 4 \, e^{t} \right\}= \frac{4}{s - 1} + e^{\left(-3 \, s\right)}\]

Example 68

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 68)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \delta\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} \right\}= -\frac{4}{s - 3} + 3 \, e^{\left(-s\right)}\]

Example 69

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 69)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \cos\left(t\right) \right\}= -\frac{3 \, s}{s^{2} + 1}\]

Example 70

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 70)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, t^{3} \right\}= \frac{12}{s^{4}}\]

Example 71

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 71)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \delta\left(t - 1\right) + 2 \, e^{t} \right\}= \frac{2}{s - 1} + e^{\left(-s\right)}\]

Example 72

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 72)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \sin\left(2 \, t\right) \right\}= \frac{8}{s^{2} + 4}\]

Example 73

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 73)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, \cos\left(3 \, t\right) \right\}= \frac{3 \, s}{s^{2} + 9}\]

Example 74

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 74)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, e^{\left(2 \, t\right)} + u\left(t - 3\right) \right\}= \frac{e^{\left(-3 \, s\right)}}{s} - \frac{3}{s - 2}\]

Example 75

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 75)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \cos\left(2 \, t\right) \right\}= -\frac{2 \, s}{s^{2} + 4}\]

Example 76

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 76)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 3 \, e^{\left(3 \, t\right)} + 3 \, u\left(t - 2\right) \right\}= \frac{3 \, e^{\left(-2 \, s\right)}}{s} + \frac{3}{s - 3}\]

Example 77

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 77)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \sin\left(t\right) \right\}= \frac{4}{s^{2} + 1}\]

Example 78

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 78)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, e^{t} + u\left(t - 1\right) \right\}= \frac{e^{\left(-s\right)}}{s} - \frac{4}{s - 1}\]

Example 79

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 79)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, \sin\left(t\right) \right\}= -\frac{2}{s^{2} + 1}\]

Example 80

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 80)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, e^{\left(3 \, t\right)} - 2 \, u\left(t - 2\right) \right\}= -\frac{2 \, e^{\left(-2 \, s\right)}}{s} + \frac{4}{s - 3}\]

Example 81

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 81)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \sin\left(t\right) \right\}= \frac{1}{s^{2} + 1}\]

Example 82

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 82)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ t^{3} \right\}= \frac{6}{s^{4}}\]

Example 83

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 83)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -t \right\}= -\frac{1}{s^{2}}\]

Example 84

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 84)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 2 \, \sin\left(2 \, t\right) \right\}= \frac{4}{s^{2} + 4}\]

Example 85

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 85)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \delta\left(t - 2\right) - 2 \, e^{\left(3 \, t\right)} \right\}= -\frac{2}{s - 3} + e^{\left(-2 \, s\right)}\]

Example 86

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 86)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \delta\left(t - 3\right) - e^{\left(3 \, t\right)} \right\}= -\frac{1}{s - 3} - 3 \, e^{\left(-3 \, s\right)}\]

Example 87

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 87)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \cos\left(t\right) \right\}= -\frac{4 \, s}{s^{2} + 1}\]

Example 88

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 88)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \delta\left(t - 3\right) + e^{\left(3 \, t\right)} \right\}= \frac{1}{s - 3} - 4 \, e^{\left(-3 \, s\right)}\]

Example 89

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 89)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ \cos\left(3 \, t\right) \right\}= \frac{s}{s^{2} + 9}\]

Example 90

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 90)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \cos\left(t\right) \right\}= -\frac{4 \, s}{s^{2} + 1}\]

Example 91

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 91)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ e^{\left(3 \, t\right)} + 2 \, u\left(t - 3\right) \right\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s} + \frac{1}{s - 3}\]

Example 92

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 92)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, t^{2} \right\}= -\frac{6}{s^{3}}\]

Example 93

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 93)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, t^{3} \right\}= -\frac{18}{s^{4}}\]

Example 94

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 94)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ 4 \, \delta\left(t - 1\right) + e^{\left(2 \, t\right)} \right\}= \frac{1}{s - 2} + 4 \, e^{\left(-s\right)}\]

Example 95

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 95)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, e^{\left(2 \, t\right)} + u\left(t - 2\right) \right\}= \frac{e^{\left(-2 \, s\right)}}{s} - \frac{4}{s - 2}\]

Example 96

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 96)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -4 \, \delta\left(t - 1\right) + 3 \, e^{\left(3 \, t\right)} \right\}= \frac{3}{s - 3} - 4 \, e^{\left(-s\right)}\]

Example 97

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 97)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, t^{3} \right\}= -\frac{12}{s^{4}}\]

Example 98

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 98)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -\cos\left(t\right) \right\}= -\frac{s}{s^{2} + 1}\]

Example 99

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 99)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -2 \, e^{\left(2 \, t\right)} - 4 \, u\left(t - 3\right) \right\}= -\frac{4 \, e^{\left(-3 \, s\right)}}{s} - \frac{2}{s - 2}\]

Example 100

D1 - Laplace transform. Compute the Laplace transform of a function from the definition. (ver. 100)

Use the definition of the Laplace transform (not a formula) to verify the following.

\[\mathcal L\left\{ -3 \, \sin\left(2 \, t\right) \right\}= -\frac{6}{s^{2} + 4}\]