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) = 13(s + 5)/(s(s + 1)(s + 3)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 13(s + 5)/(s(s + 1)(s + 3))
f(∞) =
2.Easy
t (s)f(t)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 7/(s + 2) − 9/s with a table of transform pairs, and evaluate f(0.25).

F(s) = 7/(s + 2) − 9/st = 0.25 s
f(0.25) =
3.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (5s + 17)/(s² + 12s + 35). Its inverse transform has the form f(t) = A·e−5t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (5s + 17)/(s² + 12s + 35)
A =
4.Medium
ReIm−2−4−6−8−10−12−14

Find f(t) for F(s) = (3s + 37)/(s² + 6s + 10) by completing the square, and evaluate f(0.4).

F(s) = (3s + 37)/(s² + 6s + 10)t = 0.4 s
f(0.4) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 7y′ + 12y = 48·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.2).

y″ + 7y′ + 12y = 48·u(t)y(0) = 6y′(0) = 0u(t) = 4t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

Solve y″ + 9y′ + 20y = 0 with y(0) = 2 and y′(0) = 11 using the Laplace transform, and find y(0.3).

y″ + 9y′ + 20y = 0y(0) = 2y′(0) = 11t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(∞) = 21.67
  2. 2. f(0.25) = −4.754
  3. 3. A = −4
  4. 4. f(0.4) = 4.116
  5. 5. y(0.2) = 7.527
  6. 6. y(0.3) = 2.086

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