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) = (3s + 31)/(s² + 4s + 10). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (3s + 31)/(s² + 4s + 10)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (5s² + 20s + 36)/(s(s + 2)(s + 6)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (5s² + 20s + 36)/(s(s + 2)(s + 6))
f(0⁺) =
3.Medium
ReIm−5−10−15−20

Find f(t) for F(s) = (s + 20)/(s² + 8s + 20) by completing the square, and evaluate f(0.4).

F(s) = (s + 20)/(s² + 8s + 20)t = 0.4 s
f(0.4) =
4.Medium
ReIm−5−10−15−20

Find f(t) for F(s) = (2s + 37)/(s² + 2s + 17) by completing the square, and evaluate f(0.2).

F(s) = (2s + 37)/(s² + 2s + 17)t = 0.2 s
f(0.2) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 9y′ + 20y = 20·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(0.2).

y″ + 9y′ + 20y = 20·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 7y′ + 6y = 18·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.2).

y″ + 7y′ + 6y = 18·u(t)y(0) = 6y′(0) = 0u(t) = 1t = 0.2 s
y(0.2) =

Answer key

  1. 1. f′(0⁺) = 19
  2. 2. f(0⁺) = 5
  3. 3. f(0.4) = 1.299
  4. 4. f(0.2) = 6.28
  5. 5. y(0.2) = 1.124
  6. 6. y(0.2) = 5.767

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