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) = (−s + 21)/(s² + 5s + 4). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−s + 21)/(s² + 5s + 4)
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 11(s + 6)/(s(s + 3)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 11(s + 6)/(s(s + 3)(s + 8))
f(∞) =
3.Medium
t (s)f(t)t = 0.4f(0.4) = ?

Find f(t) for F(s) = (s + 14)/(s² + 8s + 16) and evaluate f(0.4).

F(s) = (s + 14)/(s² + 8s + 16)t = 0.4 s
f(0.4) =
4.Medium
ReIm−5−10−15−20−25−30−35−40−45

Find f(t) for F(s) = (s + 40)/(s² + 4s + 8) by completing the square, and evaluate f(0.3).

F(s) = (s + 40)/(s² + 4s + 8)t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 3 and y′(0) = −11 using the Laplace transform, and find y(0.2).

y″ + 8y′ + 15y = 0y(0) = 3y′(0) = −11t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

Solve y″ + 11y′ + 30y = 0 with y(0) = 1 and y′(0) = 11 using the Laplace transform, and find y(0.1).

y″ + 11y′ + 30y = 0y(0) = 1y′(0) = 11t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(0⁺) = −1
  2. 2. f(∞) = 2.75
  3. 3. f(0.4) = 1.009
  4. 4. f(0.3) = 6.341
  5. 5. y(0.2) = 1.466
  6. 6. y(0.1) = 1.53

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