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) = (−3s + 15)/(s² + 4s + 17). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−3s + 15)/(s² + 4s + 17)
f′(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (7s + 27)/(s² + 4s + 14). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (7s + 27)/(s² + 4s + 14)
f′(0⁺) =
3.Medium
t (s)f(t)t = 1f(1) = ?

Find f(t) for F(s) = 60/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(1).

F(s) = 60/(s(s + 1)(s + 4))t = 1 s
f(1) =
4.Medium
ReIm5

Find f(t) for F(s) = 22/(s² + 4s + 40) by completing the square, and evaluate f(0.3).

F(s) = 22/(s² + 4s + 40)t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.4y(0.4) = ?

A system obeys y″ + 7y′ + 10y = 40·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.4).

y″ + 7y′ + 10y = 40·u(t)y(0) = 1y′(0) = 0u(t) = 10t = 0.4 s
y(0.4) =
6.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 3y′ + 2y = 2·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(0.5).

y″ + 3y′ + 2y = 2·u(t)y(0) = 4y′(0) = 0u(t) = 2t = 0.5 s
y(0.5) =

Answer key

  1. 1. f′(0⁺) = 27
  2. 2. f′(0⁺) = −1
  3. 3. f(1) = 7.734
  4. 4. f(0.3) = 1.96
  5. 5. y(0.4) = 14.31
  6. 6. y(0.5) = 3.69

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