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) = (7s + 40)/(s² + 3s + 25). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (7s + 40)/(s² + 3s + 25)
f′(0⁺) =
2.Easy
t (s)f(t)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 10/(s + 4) − 7/s with a table of transform pairs, and evaluate f(0.25).

F(s) = 10/(s + 4) − 7/st = 0.25 s
f(0.25) =
3.Medium
t (s)f(t)t = 1.5f(1.5) = ?

Find f(t) for F(s) = (3s + 9)/(s² + 2s + 1) and evaluate f(1.5).

F(s) = (3s + 9)/(s² + 2s + 1)t = 1.5 s
f(1.5) =
4.Medium
ReIm2

Find f(t) for F(s) = 35/(s² + 2s + 10) by completing the square, and evaluate f(0.6).

F(s) = 35/(s² + 2s + 10)t = 0.6 s
f(0.6) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 11y′ + 30y = 60·u(t), with y(0) = 4 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″ + 11y′ + 30y = 60·u(t)y(0) = 4y′(0) = 0u(t) = 2t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 8y′ + 15y = 30·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 10 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 8y′ + 15y = 30·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =

Answer key

  1. 1. f′(0⁺) = 19
  2. 2. f(0.25) = −3.321
  3. 3. f(1.5) = 2.678
  4. 4. f(0.6) = 6.235
  5. 5. y(0.2) = 4
  6. 6. y(0.3) = 6.365

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