Laplace transforms worksheet with answers

Free printable Laplace transforms worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (4s + 9)/(s² + 5s + 4). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (4s + 9)/(s² + 5s + 4)
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 20/(s(s² + 4s + 14)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 20/(s(s² + 4s + 14))
f(∞) =
3.Medium
t (s)f(t)t = 0.75f(0.75) = ?

Find f(t) for F(s) = (4s + 7)/(s² + 2s + 1) and evaluate f(0.75).

F(s) = (4s + 7)/(s² + 2s + 1)t = 0.75 s
f(0.75) =
4.Medium
ReIm−0.5−1−1.5−2−2.5−30.51

Find f(t) for F(s) = 4/(s² + 6s + 10) by completing the square, and evaluate f(0.1).

F(s) = 4/(s² + 6s + 10)t = 0.1 s
f(0.1) =
5.Hard
t (s)y(t)t = 0.4y(0.4) = ?

A system obeys y″ + 4y′ + 3y = 3·u(t), with y(0) = 2 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.4).

y″ + 4y′ + 3y = 3·u(t)y(0) = 2y′(0) = 0u(t) = 1t = 0.4 s
y(0.4) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = −9 and y′(0) = −1 using the Laplace transform, and find y(0.1).

y″ + 8y′ + 15y = 0y(0) = −9y′(0) = −1t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(0⁺) = 4
  2. 2. f(∞) = 1.429
  3. 3. f(0.75) = 2.952
  4. 4. f(0.1) = 0.2958
  5. 5. y(0.4) = 1.855
  6. 6. y(0.1) = −8.547

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