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 a table of Laplace transformations to find the following.

\[L\left\{3 \, t^{3} - \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{18}{s^{4}} - e^{\left(-3 \, s\right)}\]


Example 2

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, \cos\left(3 \, t\right) - 2 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{2 \, s}{s^{2} + 9} - \frac{2 \, e^{\left(-s\right)}}{s}\]


Example 3

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \cos\left(3 \, t\right) + 2 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, s}{s^{2} + 9} + 2 \, e^{\left(-3 \, s\right)}\]


Example 4

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, \sin\left(2 \, t\right) - 3 \, u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{3 \, e^{\left(-3 \, s\right)}}{s} + \frac{4}{s^{2} + 4}\]


Example 5

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, \delta\left(t - 2\right) - 3 \, \sin\left(2 \, t\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{6}{s^{2} + 4} - 2 \, e^{\left(-2 \, s\right)}\]


Example 6

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

Use a table of Laplace transformations to find the following.

\[L\left\{4 \, \cos\left(2 \, t\right) - 2 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4 \, s}{s^{2} + 4} - 2 \, e^{\left(-3 \, s\right)}\]


Example 7

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

Use a table of Laplace transformations to find the following.

\[L\left\{4 \, \cos\left(3 \, t\right) + 4 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4 \, s}{s^{2} + 9} + 4 \, e^{\left(-3 \, s\right)}\]


Example 8

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

Use a table of Laplace transformations to find the following.

\[L\left\{4 \, \sin\left(3 \, t\right) + u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{e^{\left(-3 \, s\right)}}{s} + \frac{12}{s^{2} + 9}\]


Example 9

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, \sin\left(2 \, t\right) + 2 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{2 \, e^{\left(-s\right)}}{s} + \frac{4}{s^{2} + 4}\]


Example 10

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, t^{2} - u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{e^{\left(-3 \, s\right)}}{s} + \frac{4}{s^{3}}\]


Example 11

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, t^{2} - u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{e^{\left(-s\right)}}{s} - \frac{4}{s^{3}}\]


Example 12

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

Use a table of Laplace transformations to find the following.

\[L\left\{-3 \, t^{3} - 4 \, u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, e^{\left(-2 \, s\right)}}{s} - \frac{18}{s^{4}}\]


Example 13

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, t^{2} - 3 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{8}{s^{3}} - 3 \, e^{\left(-s\right)}\]


Example 14

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

Use a table of Laplace transformations to find the following.

\[L\left\{-t^{3} + u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{e^{\left(-2 \, s\right)}}{s} - \frac{6}{s^{4}}\]


Example 15

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, \cos\left(2 \, t\right) - 3 \, \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{3 \, s}{s^{2} + 4} - 3 \, e^{\left(-2 \, s\right)}\]


Example 16

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

Use a table of Laplace transformations to find the following.

\[L\left\{-3 \, t^{3} - 4 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{18}{s^{4}} - 4 \, e^{\left(-s\right)}\]


Example 17

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, t^{3} - 4 \, u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, e^{\left(-2 \, s\right)}}{s} - \frac{12}{s^{4}}\]


Example 18

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, \cos\left(2 \, t\right) + 2 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{2 \, s}{s^{2} + 4} + 2 \, e^{\left(-3 \, s\right)}\]


Example 19

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \sin\left(2 \, t\right) + 3 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{3 \, e^{\left(-s\right)}}{s} - \frac{8}{s^{2} + 4}\]


Example 20

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

Use a table of Laplace transformations to find the following.

\[L\left\{-3 \, t^{2} + 4 \, u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4 \, e^{\left(-3 \, s\right)}}{s} - \frac{6}{s^{3}}\]


Example 21

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

Use a table of Laplace transformations to find the following.

\[L\left\{-\sin\left(2 \, t\right) - 3 \, u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{3 \, e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 4}\]


Example 22

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, \sin\left(3 \, t\right) - u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{e^{\left(-3 \, s\right)}}{s} - \frac{6}{s^{2} + 9}\]


Example 23

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

Use a table of Laplace transformations to find the following.

\[L\left\{-\sin\left(2 \, t\right) - 2 \, u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{2 \, e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 4}\]


Example 24

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, t^{2} - 3 \, u\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{3 \, e^{\left(-3 \, s\right)}}{s} - \frac{8}{s^{3}}\]


Example 25

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, t^{3} + 3 \, u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{3 \, e^{\left(-2 \, s\right)}}{s} - \frac{24}{s^{4}}\]


Example 26

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, \cos\left(3 \, t\right) - 3 \, u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{3 \, s}{s^{2} + 9} - \frac{3 \, e^{\left(-2 \, s\right)}}{s}\]


Example 27

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

Use a table of Laplace transformations to find the following.

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

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{24}{s^{4}} + 2 \, e^{\left(-s\right)}\]


Example 28

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \cos\left(3 \, t\right) + 2 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, s}{s^{2} + 9} + 2 \, e^{\left(-s\right)}\]


Example 29

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, t^{3} - u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{e^{\left(-2 \, s\right)}}{s} + \frac{18}{s^{4}}\]


Example 30

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \cos\left(2 \, t\right) - 3 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, s}{s^{2} + 4} - 3 \, e^{\left(-3 \, s\right)}\]


Example 31

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

Use a table of Laplace transformations to find the following.

\[L\left\{\delta\left(t - 3\right) - 3 \, \sin\left(2 \, t\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{6}{s^{2} + 4} + e^{\left(-3 \, s\right)}\]


Example 32

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, \sin\left(3 \, t\right) + 4 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4 \, e^{\left(-s\right)}}{s} + \frac{9}{s^{2} + 9}\]


Example 33

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, \sin\left(3 \, t\right) + 4 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4 \, e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}\]


Example 34

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, \delta\left(t - 3\right) + 4 \, \sin\left(3 \, t\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{12}{s^{2} + 9} - 2 \, e^{\left(-3 \, s\right)}\]


Example 35

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \cos\left(3 \, t\right) + 4 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, s}{s^{2} + 9} + 4 \, e^{\left(-3 \, s\right)}\]


Example 36

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

Use a table of Laplace transformations to find the following.

\[L\left\{\delta\left(t - 2\right) + 2 \, \sin\left(2 \, t\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4}{s^{2} + 4} + e^{\left(-2 \, s\right)}\]


Example 37

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, t^{3} + 2 \, \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{18}{s^{4}} + 2 \, e^{\left(-2 \, s\right)}\]


Example 38

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

Use a table of Laplace transformations to find the following.

\[L\left\{-t^{3} - 3 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{6}{s^{4}} - 3 \, e^{\left(-s\right)}\]


Example 39

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

Use a table of Laplace transformations to find the following.

\[L\left\{\cos\left(2 \, t\right) + 2 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{s}{s^{2} + 4} + \frac{2 \, e^{\left(-s\right)}}{s}\]


Example 40

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

Use a table of Laplace transformations to find the following.

\[L\left\{-2 \, t^{3} - \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{12}{s^{4}} - e^{\left(-2 \, s\right)}\]


Example 41

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, \cos\left(3 \, t\right) + \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{4 \, s}{s^{2} + 9} + e^{\left(-2 \, s\right)}\]


Example 42

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, t^{2} + 3 \, \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{8}{s^{3}} + 3 \, e^{\left(-2 \, s\right)}\]


Example 43

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

Use a table of Laplace transformations to find the following.

\[L\left\{-3 \, \cos\left(3 \, t\right) - 3 \, u\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{3 \, s}{s^{2} + 9} - \frac{3 \, e^{\left(-2 \, s\right)}}{s}\]


Example 44

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

Use a table of Laplace transformations to find the following.

\[L\left\{-\cos\left(2 \, t\right) - \delta\left(t - 2\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{s}{s^{2} + 4} - e^{\left(-2 \, s\right)}\]


Example 45

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, t^{2} - 3 \, u\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{3 \, e^{\left(-s\right)}}{s} + \frac{6}{s^{3}}\]


Example 46

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

Use a table of Laplace transformations to find the following.

\[L\left\{-4 \, t^{2} - 2 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{8}{s^{3}} - 2 \, e^{\left(-3 \, s\right)}\]


Example 47

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

Use a table of Laplace transformations to find the following.

\[L\left\{2 \, t^{2} - 2 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{4}{s^{3}} - 2 \, e^{\left(-s\right)}\]


Example 48

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

Use a table of Laplace transformations to find the following.

\[L\left\{-3 \, t^{3} + 4 \, \delta\left(t - 3\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{18}{s^{4}} + 4 \, e^{\left(-3 \, s\right)}\]


Example 49

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

Use a table of Laplace transformations to find the following.

\[L\left\{-\cos\left(3 \, t\right) - 2 \, \delta\left(t - 1\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[-\frac{s}{s^{2} + 9} - 2 \, e^{\left(-s\right)}\]


Example 50

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

Use a table of Laplace transformations to find the following.

\[L\left\{3 \, \delta\left(t - 2\right) + \sin\left(2 \, t\right)\right\}\]

Then verify this result by using the integral definition of a Laplace transform. (You may use a table of integrals.)

Answer.

\[\frac{2}{s^{2} + 4} + 3 \, e^{\left(-2 \, s\right)}\]