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)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−6s² + 13s + 19)/(s(s + 5)(s + 6)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−6s² + 13s + 19)/(s(s + 5)(s + 6))
f(0⁺) =
2.Easy
t (s)f(t)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 12/(s² + 9) with a table of transform pairs, and evaluate f(0.1).

F(s) = 12/(s² + 9)t = 0.1 s
f(0.1) =
3.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (6s + 38)/(s² + 6s + 9) and evaluate f(0.25).

F(s) = (6s + 38)/(s² + 6s + 9)t = 0.25 s
f(0.25) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (5s + 31)/(s² + 13s + 42). Its inverse transform has the form f(t) = A·e−6t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (5s + 31)/(s² + 13s + 42)
A =
5.Hard
t (s)y(t)t = 1y(1) = ?

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

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

A system obeys y″ + 8y′ + 12y = 12·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.4).

y″ + 8y′ + 12y = 12·u(t)y(0) = 6y′(0) = 0u(t) = 10t = 0.4 s
y(0.4) =

Answer key

  1. 1. f(0⁺) = −6
  2. 2. f(0.1) = 1.182
  3. 3. f(0.25) = 5.196
  4. 4. A = 1
  5. 5. y(1) = 3.201
  6. 6. y(0.4) = 7.485

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