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) = (−s + 34)/(s² + 11s + 28). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−s + 34)/(s² + 11s + 28)
f(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−6s + 40)/(s² + 2s + 11). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−6s + 40)/(s² + 2s + 11)
f′(0⁺) =
3.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (5s + 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) = (5s + 32)/(s² + 13s + 42)
A =
4.Medium
t (s)f(t)t = 1f(1) = ?

Find f(t) for F(s) = (6s + 15)/(s² + 2s + 1) and evaluate f(1).

F(s) = (6s + 15)/(s² + 2s + 1)t = 1 s
f(1) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 10y′ + 24y = 60·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 10 for t ≥ 0. Use the Laplace transform to find the output y(0.2).

y″ + 10y′ + 24y = 60·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 4y′ + 3y = 12·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(0.2).

y″ + 4y′ + 3y = 12·u(t)y(0) = 6y′(0) = 0u(t) = 1t = 0.2 s
y(0.2) =

Answer key

  1. 1. f(0⁺) = −1
  2. 2. f′(0⁺) = 52
  3. 3. A = 2
  4. 4. f(1) = 5.518
  5. 5. y(0.2) = 6.36
  6. 6. y(0.2) = 5.907

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