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(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−4s² + 13s + 21)/(s(s + 7)(s + 8)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−4s² + 13s + 21)/(s(s + 7)(s + 8))
f(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−2s + 22)/(s² + 8s + 9). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−2s + 22)/(s² + 8s + 9)
f′(0⁺) =
3.Medium
ReIm2

Find f(t) for F(s) = (3s + 4)/(s² + 2s + 10) by completing the square, and evaluate f(0.4).

F(s) = (3s + 4)/(s² + 2s + 10)t = 0.4 s
f(0.4) =
4.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = 35/(s(s + 2)(s + 6)) by partial fractions, and evaluate f(0.5).

F(s) = 35/(s(s + 2)(s + 6))t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 8y′ + 12y = 12·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.5).

y″ + 8y′ + 12y = 12·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.5 s
y(0.5) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 9y′ + 20y = 60·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.1).

y″ + 9y′ + 20y = 60·u(t)y(0) = 1y′(0) = 0u(t) = 1t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(0⁺) = −4
  2. 2. f′(0⁺) = 38
  3. 3. f(0.4) = 0.9369
  4. 4. f(0.5) = 1.38
  5. 5. y(0.5) = 0.4731
  6. 6. y(0.1) = 1.149

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