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) = 2(s + 3)/(s(s + 2)(s + 6)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 2(s + 3)/(s(s + 2)(s + 6))
f(∞) =
2.Easy
t (s)f(t)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 12/(s² + 25) with a table of transform pairs, and evaluate f(0.25).

F(s) = 12/(s² + 25)t = 0.25 s
f(0.25) =
3.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = (2s + 40)/(s² + 2s + 1) and evaluate f(0.3).

F(s) = (2s + 40)/(s² + 2s + 1)t = 0.3 s
f(0.3) =
4.Medium
ReIm−2−4−6−8−10

F(s) = (2s + 18)/(s² + 9s + 14). Its inverse transform has the form f(t) = A·e−2t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (2s + 18)/(s² + 9s + 14)
A =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 11y′ + 30y = 60·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(0.1).

y″ + 11y′ + 30y = 60·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 8y′ + 15y = 45·u(t), with y(0) = 2 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.1).

y″ + 8y′ + 15y = 45·u(t)y(0) = 2y′(0) = 0u(t) = 10t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(∞) = 0.5
  2. 2. f(0.25) = 2.278
  3. 3. f(0.3) = 9.927
  4. 4. A = 2.8
  5. 5. y(0.1) = 2.097
  6. 6. y(0.1) = 3.617

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