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) = 12(s + 1)/(s(s + 3)(s + 5)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 12(s + 1)/(s(s + 3)(s + 5))
f(∞) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (s + 39)/(s² + 14s + 48). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (s + 39)/(s² + 14s + 48)
f(0⁺) =
3.Medium
t (s)f(t)t = 0.8f(0.8) = ?

Find f(t) for F(s) = 35/(s(s + 1)(s + 5)) by partial fractions, and evaluate f(0.8).

F(s) = 35/(s(s + 1)(s + 5))t = 0.8 s
f(0.8) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (6s + 12)/(s² + 13s + 42). Its inverse transform has the form f(t) = A·e−6t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (6s + 12)/(s² + 13s + 42)
A =
5.Hard
t (s)y(t)t = 0.6y(0.6) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 5 and y′(0) = −4 using the Laplace transform, and find y(0.6).

y″ + 3y′ + 2y = 0y(0) = 5y′(0) = −4t = 0.6 s
y(0.6) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 11y′ + 30y = 120·u(t), with y(0) = 5 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.1).

y″ + 11y′ + 30y = 120·u(t)y(0) = 5y′(0) = 0u(t) = 2t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(∞) = 0.8
  2. 2. f(0⁺) = 1
  3. 3. f(0.8) = 3.1
  4. 4. A = −24
  5. 5. y(0.6) = 2.992
  6. 6. y(0.1) = 5.315

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