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 + 4)/(s² + 7s + 3). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (s + 4)/(s² + 7s + 3)
f′(0⁺) =
2.Easy
t (s)f(t)t = 0.3f(0.3) = ?

Find the inverse Laplace transform f(t) of F(s) = 8/(s + 1) − 3/s with a table of transform pairs, and evaluate f(0.3).

F(s) = 8/(s + 1) − 3/st = 0.3 s
f(0.3) =
3.Medium
ReIm−55

Find f(t) for F(s) = (4s + 28)/(s² + 6s + 45) by completing the square, and evaluate f(0.6).

F(s) = (4s + 28)/(s² + 6s + 45)t = 0.6 s
f(0.6) =
4.Medium
ReIm−2−4−6−8−10−1224

Find f(t) for F(s) = (3s + 32)/(s² + 8s + 32) by completing the square, and evaluate f(0.4).

F(s) = (3s + 32)/(s² + 8s + 32)t = 0.4 s
f(0.4) =
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) = 2 for t ≥ 0. Use the Laplace transform to find the output y(0.1).

y″ + 7y′ + 12y = 48·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 9y′ + 18y = 36·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.2).

y″ + 9y′ + 18y = 36·u(t)y(0) = 6y′(0) = 0u(t) = 2t = 0.2 s
y(0.2) =

Answer key

  1. 1. f′(0⁺) = −3
  2. 2. f(0.3) = 2.927
  3. 3. f(0.6) = −0.788
  4. 4. f(0.4) = 0.9914
  5. 5. y(0.1) = 0.3815
  6. 6. y(0.2) = 5.593

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