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) = (s + 15)/(s² + 9s + 13). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (s + 15)/(s² + 9s + 13)
f′(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 9(s + 2)/(s(s + 3)(s + 6)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 9(s + 2)/(s(s + 3)(s + 6))
f(∞) =
3.Medium
ReIm−1−2−3−4−5−6−7−812

Find f(t) for F(s) = (3s + 22)/(s² + 4s + 8) by completing the square, and evaluate f(0.2).

F(s) = (3s + 22)/(s² + 4s + 8)t = 0.2 s
f(0.2) =
4.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (s + 21)/(s² + 10s + 25) and evaluate f(0.25).

F(s) = (s + 21)/(s² + 10s + 25)t = 0.25 s
f(0.25) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 7y′ + 10y = 50·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 2 for t ≥ 0. Use the Laplace transform to find the output y(0.2).

y″ + 7y′ + 10y = 50·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.4y(0.4) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 1 and y′(0) = −9 using the Laplace transform, and find y(0.4).

y″ + 4y′ + 3y = 0y(0) = 1y′(0) = −9t = 0.4 s
y(0.4) =

Answer key

  1. 1. f′(0⁺) = 6
  2. 2. f(∞) = 1
  3. 3. f(0.2) = 3.94
  4. 4. f(0.25) = 1.433
  5. 5. y(0.2) = 1.281
  6. 6. y(0.4) = −0.8062

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