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) = (4s² + 7s + 47)/(s(s + 4)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (4s² + 7s + 47)/(s(s + 4)(s + 7))
f(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (2s + 17)/(s² + 11s + 28). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (2s + 17)/(s² + 11s + 28)
f(0⁺) =
3.Medium
ReIm−2−4−6−8

F(s) = (4s + 34)/(s² + 4s + 3). Its inverse transform has the form f(t) = A·e−t + B·e−3t. Find the coefficient A by partial fractions.

F(s) = (4s + 34)/(s² + 4s + 3)
A =
4.Medium
t (s)f(t)t = 0.8f(0.8) = ?

Find f(t) for F(s) = 30/(s(s + 1)(s + 6)) by partial fractions, and evaluate f(0.8).

F(s) = 30/(s(s + 1)(s + 6))t = 0.8 s
f(0.8) =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

Solve y″ + 7y′ + 6y = 0 with y(0) = 5 and y′(0) = 3 using the Laplace transform, and find y(0.3).

y″ + 7y′ + 6y = 0y(0) = 5y′(0) = 3t = 0.3 s
y(0.3) =
6.Hard
ReIm−2−4−6−8

A system is described by the differential equation y″ + 3y′ + 2y = 2u′ + 17u, 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 = 1 s. Find g(1).

y″ + 3y′ + 2y = 2u′ + 17ut = 1 s
g(1) =

Answer key

  1. 1. f(0⁺) = 4
  2. 2. f(0⁺) = 2
  3. 3. A = 15
  4. 4. f(0.8) = 2.312
  5. 5. y(0.3) = 4.625
  6. 6. g(1) = 3.759

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