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)

A signal f(t) has the Laplace transform F(s) = (6s + 4)/(s² + 3s + 8). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (6s + 4)/(s² + 3s + 8)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−9s + 40)/(s² + 3s + 2). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−9s + 40)/(s² + 3s + 2)
f(0⁺) =
3.Medium
ReIm−5−105

Find f(t) for F(s) = (2s + 26)/(s² + 8s + 52) by completing the square, and evaluate f(0.2).

F(s) = (2s + 26)/(s² + 8s + 52)t = 0.2 s
f(0.2) =
4.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = (4s + 6)/(s² + 4s + 4) and evaluate f(0.5).

F(s) = (4s + 6)/(s² + 4s + 4)t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 7y′ + 12y = 48·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(0.1).

y″ + 7y′ + 12y = 48·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 7y′ + 10y = 30·u(t), with y(0) = 4 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.25).

y″ + 7y′ + 10y = 30·u(t)y(0) = 4y′(0) = 0u(t) = 2t = 0.25 s
y(0.25) =

Answer key

  1. 1. f′(0⁺) = −14
  2. 2. f(0⁺) = −9
  3. 3. f(0.2) = 1.582
  4. 4. f(0.5) = 1.104
  5. 5. y(0.1) = 0.9537
  6. 6. y(0.25) = 4.36

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