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(∞) = ?

A signal f(t) has the Laplace transform F(s) = 40/(s(s² + 3s + 30)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 40/(s(s² + 3s + 30))
f(∞) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−s + 11)/(s² + 4s + 28). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−s + 11)/(s² + 4s + 28)
f′(0⁺) =
3.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = 40/(s(s + 2)(s + 6)) by partial fractions, and evaluate f(0.2).

F(s) = 40/(s(s + 2)(s + 6))t = 0.2 s
f(0.2) =
4.Medium
ReIm−2−4−6−8−10−12−14

F(s) = (3s + 38)/(s² + 8s + 12). Its inverse transform has the form f(t) = A·e−2t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (3s + 38)/(s² + 8s + 12)
A =
5.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 4y′ + 3y = 9·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(0.5).

y″ + 4y′ + 3y = 9·u(t)y(0) = 1y′(0) = 0u(t) = 10t = 0.5 s
y(0.5) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 10y′ + 24y = 72·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 5 is applied. Use the Laplace transform to find y(0.3).

y″ + 10y′ + 24y = 72·u(t)y(0) = 6y′(0) = 0u(t) = 5t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(∞) = 1.333
  2. 2. f′(0⁺) = 15
  3. 3. f(0.2) = 0.4837
  4. 4. A = 8
  5. 5. y(0.5) = 6.851
  6. 6. y(0.3) = 9.843

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