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) = 20(s + 4)/(s(s + 6)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 20(s + 4)/(s(s + 6)(s + 7))
f(∞) =
2.Easy
t (s)f(t)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 9/(s + 5)² with a table of transform pairs, and evaluate f(0.1).

F(s) = 9/(s + 5)²t = 0.1 s
f(0.1) =
3.Medium
ReIm−2−4−6−8−10−12

F(s) = (3s + 32)/(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) = (3s + 32)/(s² + 13s + 42)
A =
4.Medium
ReIm−1−2−3−4−5−6−71

Find f(t) for F(s) = (4s + 27)/(s² + 6s + 10) by completing the square, and evaluate f(0.6).

F(s) = (4s + 27)/(s² + 6s + 10)t = 0.6 s
f(0.6) =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 3y′ + 2y = 3·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 5 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 3y′ + 2y = 3·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 7y′ + 6y = 6·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.25).

y″ + 7y′ + 6y = 6·u(t)y(0) = 6y′(0) = 0u(t) = 1t = 0.25 s
y(0.25) =

Answer key

  1. 1. f(∞) = 1.905
  2. 2. f(0.1) = 0.5459
  3. 3. A = 14
  4. 4. f(0.6) = 1.946
  5. 5. y(0.3) = 0.5038
  6. 6. y(0.25) = 5.45

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