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) = (4s + 28)/(s² + 6s + 3). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (4s + 28)/(s² + 6s + 3)
f′(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 20/(s(s² + 8s + 24)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 20/(s(s² + 8s + 24))
f(∞) =
3.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = (4s + 17)/(s² + 8s + 16) and evaluate f(0.2).

F(s) = (4s + 17)/(s² + 8s + 16)t = 0.2 s
f(0.2) =
4.Medium
t (s)f(t)t = 1.5f(1.5) = ?

Find f(t) for F(s) = 55/(s(s + 1)(s + 3)) by partial fractions, and evaluate f(1.5).

F(s) = 55/(s(s + 1)(s + 3))t = 1.5 s
f(1.5) =
5.Hard
t (s)y(t)t = 1y(1) = ?

A system obeys y″ + 5y′ + 4y = 12·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(1).

y″ + 5y′ + 4y = 12·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 1 s
y(1) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 9 and y′(0) = 0 using the Laplace transform, and find y(0.6).

y″ + 7y′ + 10y = 0y(0) = 9y′(0) = 0t = 0.6 s
y(0.6) =

Answer key

  1. 1. f′(0⁺) = 4
  2. 2. f(∞) = 0.8333
  3. 3. f(0.2) = 1.887
  4. 4. f(1.5) = 12.3
  5. 5. y(1) = 15.47
  6. 6. y(0.6) = 4.219

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