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(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (5s² + 19s + 50)/(s(s + 4)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (5s² + 19s + 50)/(s(s + 4)(s + 7))
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 15(s + 2)/(s(s + 1)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 15(s + 2)/(s(s + 1)(s + 7))
f(∞) =
3.Medium
ReIm−2−4−6−8

F(s) = (2s + 17)/(s² + 7s + 10). Its inverse transform has the form f(t) = A·e−2t + B·e−5t. Find the coefficient A by partial fractions.

F(s) = (2s + 17)/(s² + 7s + 10)
A =
4.Medium
ReIm−2−4−6−8−10−12

F(s) = (3s + 36)/(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) = (3s + 36)/(s² + 5s + 4)
A =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

Solve y″ + 8y′ + 12y = 0 with y(0) = 5 and y′(0) = 3 using the Laplace transform, and find y(0.1).

y″ + 8y′ + 12y = 0y(0) = 5y′(0) = 3t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.4y(0.4) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 9 and y′(0) = 3 using the Laplace transform, and find y(0.4).

y″ + 7y′ + 10y = 0y(0) = 9y′(0) = 3t = 0.4 s
y(0.4) =

Answer key

  1. 1. f(0⁺) = 5
  2. 2. f(∞) = 4.286
  3. 3. A = 4.333
  4. 4. A = 11
  5. 5. y(0.1) = 4.971
  6. 6. y(0.4) = 6.242

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