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) = (9s² + 25s + 60)/(s(s + 1)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (9s² + 25s + 60)/(s(s + 1)(s + 7))
f(0⁺) =
2.Easy
t (s)f(t)t = 1f(1) = ?

Find the inverse Laplace transform f(t) of F(s) = 3s/(s² + 4) with a table of transform pairs, and evaluate f(1).

F(s) = 3s/(s² + 4)t = 1 s
f(1) =
3.Medium
ReIm−2−4−6−8−10−12

F(s) = (2s + 24)/(s² + 10s + 21). Its inverse transform has the form f(t) = A·e−3t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (2s + 24)/(s² + 10s + 21)
A =
4.Medium
ReIm−2−42

Find f(t) for F(s) = (4s + 20)/(s² + 6s + 18) by completing the square, and evaluate f(0.4).

F(s) = (4s + 20)/(s² + 6s + 18)t = 0.4 s
f(0.4) =
5.Hard
t (s)y(t)t = 0.75y(0.75) = ?

A system obeys y″ + 6y′ + 5y = 5·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(0.75).

y″ + 6y′ + 5y = 5·u(t)y(0) = 6y′(0) = 0u(t) = 1t = 0.75 s
y(0.75) =
6.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 6y′ + 8y = 8·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.5).

y″ + 6y′ + 8y = 8·u(t)y(0) = 1y′(0) = 0u(t) = 4t = 0.5 s
y(0.5) =

Answer key

  1. 1. f(0⁺) = 9
  2. 2. f(1) = −1.248
  3. 3. A = 4.5
  4. 4. f(0.4) = 1.185
  5. 5. y(0.75) = 3.923
  6. 6. y(0.5) = 2.199

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