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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−4s + 38)/(s² + 8s + 3). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−4s + 38)/(s² + 8s + 3)
f′(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 30/(s(s² + 2s + 25)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 30/(s(s² + 2s + 25))
f(∞) =
3.Medium
ReIm−2−424

Find f(t) for F(s) = 5/(s² + 8s + 32) by completing the square, and evaluate f(0.3).

F(s) = 5/(s² + 8s + 32)t = 0.3 s
f(0.3) =
4.Medium
ReIm5

Find f(t) for F(s) = (4s + 17)/(s² + 4s + 40) by completing the square, and evaluate f(0.6).

F(s) = (4s + 17)/(s² + 4s + 40)t = 0.6 s
f(0.6) =
5.Hard
ReIm−55

A system is described by the differential equation y″ + 4y′ + 29y = 2u′ + 13u, with input u(t) and output y(t). Find its transfer function G(s) = Y(s)/U(s) (zero initial conditions) and evaluate the impulse response g(t) at t = 0.75 s. Find g(0.75).

y″ + 4y′ + 29y = 2u′ + 13ut = 0.75 s
g(0.75) =
6.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 7y′ + 10y = 40·u(t), with y(0) = 4 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.5).

y″ + 7y′ + 10y = 40·u(t)y(0) = 4y′(0) = 0u(t) = 1t = 0.5 s
y(0.5) =

Answer key

  1. 1. f′(0⁺) = 70
  2. 2. f(∞) = 1.2
  3. 3. f(0.3) = 0.3509
  4. 4. f(0.6) = −1.28
  5. 5. g(0.75) = −0.5957
  6. 6. y(0.5) = 4

Full step-by-step solutions: practise this topic on examproblems.com.