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) = 19(s + 5)/(s(s + 3)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 19(s + 5)/(s(s + 3)(s + 7))
f(∞) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 18/(s(s² + 4s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 18/(s(s² + 4s + 7))
f(∞) =
3.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = (s + 6)/(s² + 10s + 25) and evaluate f(0.1).

F(s) = (s + 6)/(s² + 10s + 25)t = 0.1 s
f(0.1) =
4.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 50/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.3).

F(s) = 50/(s(s + 4)(s + 5))t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.5y(0.5) = ?

Solve y″ + 8y′ + 12y = 0 with y(0) = 9 and y′(0) = −1 using the Laplace transform, and find y(0.5).

y″ + 8y′ + 12y = 0y(0) = 9y′(0) = −1t = 0.5 s
y(0.5) =
6.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 9y′ + 18y = 108·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 2 for t ≥ 0. Use the Laplace transform to find the output y(0.2).

y″ + 9y′ + 18y = 108·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =

Answer key

  1. 1. f(∞) = 4.524
  2. 2. f(∞) = 2.571
  3. 3. f(0.1) = 0.6672
  4. 4. f(0.3) = 0.9664
  5. 5. y(0.5) = 4.663
  6. 6. y(0.2) = 2.443

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