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) = (5s + 37)/(s² + 9s + 29). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (5s + 37)/(s² + 9s + 29)
f′(0⁺) =
2.Easy
t (s)f(t)t = 1.2f(1.2) = ?

Find the inverse Laplace transform f(t) of F(s) = 7/(s² + 16) with a table of transform pairs, and evaluate f(1.2).

F(s) = 7/(s² + 16)t = 1.2 s
f(1.2) =
3.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = (s + 13)/(s² + 10s + 25) and evaluate f(0.2).

F(s) = (s + 13)/(s² + 10s + 25)t = 0.2 s
f(0.2) =
4.Medium
ReIm−5−10−155

Find f(t) for F(s) = (3s + 40)/(s² + 8s + 52) by completing the square, and evaluate f(0.25).

F(s) = (3s + 40)/(s² + 8s + 52)t = 0.25 s
f(0.25) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 9y′ + 18y = 90·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′ + 18y = 90·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.4y(0.4) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 9 and y′(0) = 14 using the Laplace transform, and find y(0.4).

y″ + 5y′ + 6y = 0y(0) = 9y′(0) = 14t = 0.4 s
y(0.4) =

Answer key

  1. 1. f′(0⁺) = −8
  2. 2. f(1.2) = −1.743
  3. 3. f(0.2) = 0.9565
  4. 4. f(0.25) = 1.791
  5. 5. y(0.2) = 5.089
  6. 6. y(0.4) = 8.784

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