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(∞) = ?

A signal f(t) has the Laplace transform F(s) = 13/(s(s² + 2s + 27)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 13/(s(s² + 2s + 27))
f(∞) =
2.Easy
t (s)f(t)t = 0.6f(0.6) = ?

Find the inverse Laplace transform f(t) of F(s) = 12s/(s² + 25) with a table of transform pairs, and evaluate f(0.6).

F(s) = 12s/(s² + 25)t = 0.6 s
f(0.6) =
3.Medium
t (s)f(t)t = 0.6f(0.6) = ?

Find f(t) for F(s) = (6s + 40)/(s² + 4s + 4) and evaluate f(0.6).

F(s) = (6s + 40)/(s² + 4s + 4)t = 0.6 s
f(0.6) =
4.Medium
t (s)f(t)t = 0.6f(0.6) = ?

Find f(t) for F(s) = (s + 38)/(s² + 4s + 4) and evaluate f(0.6).

F(s) = (s + 38)/(s² + 4s + 4)t = 0.6 s
f(0.6) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 10 using the Laplace transform, and find y(0.2).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 10t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.75y(0.75) = ?

A system obeys y″ + 6y′ + 5y = 7.5·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 5 for t ≥ 0. Use the Laplace transform to find the output y(0.75).

y″ + 6y′ + 5y = 7.5·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.75 s
y(0.75) =

Answer key

  1. 1. f(∞) = 0.4815
  2. 2. f(0.6) = −11.88
  3. 3. f(0.6) = 6.867
  4. 4. f(0.6) = 6.807
  5. 5. y(0.2) = 3.537
  6. 6. y(0.75) = 3.116

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