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) = 4(s + 2)/(s(s + 1)(s + 4)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 4(s + 2)/(s(s + 1)(s + 4))
f(∞) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 8/(s(s² + 8s + 23)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 8/(s(s² + 8s + 23))
f(∞) =
3.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = (3s + 27)/(s² + 4s + 4) and evaluate f(0.5).

F(s) = (3s + 27)/(s² + 4s + 4)t = 0.5 s
f(0.5) =
4.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = 35/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(0.5).

F(s) = 35/(s(s + 1)(s + 4))t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 8y′ + 15y = 60·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(0.25).

y″ + 8y′ + 15y = 60·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.25 s
y(0.25) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 8y′ + 15y = 30·u(t), with y(0) = 3 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.3).

y″ + 8y′ + 15y = 30·u(t)y(0) = 3y′(0) = 0u(t) = 1t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(∞) = 2
  2. 2. f(∞) = 0.3478
  3. 3. f(0.5) = 4.966
  4. 4. f(0.5) = 2.069
  5. 5. y(0.25) = 1.991
  6. 6. y(0.3) = 2.682

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