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) = 9(s + 7)/(s(s + 1)(s + 6)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 9(s + 7)/(s(s + 1)(s + 6))
f(∞) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 18/(s(s² + 9s + 18)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 18/(s(s² + 9s + 18))
f(∞) =
3.Medium
ReIm−2−4−6−8−1024

Find f(t) for F(s) = (3s + 27)/(s² + 2s + 17) by completing the square, and evaluate f(0.6).

F(s) = (3s + 27)/(s² + 2s + 17)t = 0.6 s
f(0.6) =
4.Medium
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

F(s) = 25/(s² + 5s + 4). Its inverse transform has the form f(t) = A·e−t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = 25/(s² + 5s + 4)
A =
5.Hard
t (s)y(t)t = 0.6y(0.6) = ?

Solve y″ + 7y′ + 6y = 0 with y(0) = 5 and y′(0) = −8 using the Laplace transform, and find y(0.6).

y″ + 7y′ + 6y = 0y(0) = 5y′(0) = −8t = 0.6 s
y(0.6) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 7y′ + 12y = 12·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(0.25).

y″ + 7y′ + 12y = 12·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.25 s
y(0.25) =

Answer key

  1. 1. f(∞) = 10.5
  2. 2. f(∞) = 1
  3. 3. f(0.6) = 1.01
  4. 4. A = 8.333
  5. 5. y(0.6) = 2.431
  6. 6. y(0.25) = 0.4283

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