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) = (−6s + 25)/(s² + 6s + 8). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−6s + 25)/(s² + 6s + 8)
f(0⁺) =
2.Easy
t (s)f(t)t = 1f(1) = ?

Find the inverse Laplace transform f(t) of F(s) = 8/(s + 2)² with a table of transform pairs, and evaluate f(1).

F(s) = 8/(s + 2)²t = 1 s
f(1) =
3.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = (6s + 38)/(s² + 8s + 16) and evaluate f(0.3).

F(s) = (6s + 38)/(s² + 8s + 16)t = 0.3 s
f(0.3) =
4.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = (4s + 11)/(s² + 8s + 16) and evaluate f(0.5).

F(s) = (4s + 11)/(s² + 8s + 16)t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 10y′ + 24y = 24·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.2).

y″ + 10y′ + 24y = 24·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

A system obeys y″ + 7y′ + 6y = 6·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(0.6).

y″ + 7y′ + 6y = 6·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.6 s
y(0.6) =

Answer key

  1. 1. f(0⁺) = −6
  2. 2. f(1) = 1.083
  3. 3. f(0.3) = 3.072
  4. 4. f(0.5) = 0.203
  5. 5. y(0.2) = 0.2544
  6. 6. y(0.6) = 0.6938

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