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)

A signal f(t) has the Laplace transform F(s) = (8s + 20)/(s² + 5s + 23). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (8s + 20)/(s² + 5s + 23)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−7s² + 10s + 52)/(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) = (−7s² + 10s + 52)/(s(s + 1)(s + 7))
f(0⁺) =
3.Medium
t (s)f(t)t = 1f(1) = ?

Find f(t) for F(s) = 35/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(1).

F(s) = 35/(s(s + 1)(s + 4))t = 1 s
f(1) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (4s + 11)/(s² + 11s + 28). Its inverse transform has the form f(t) = A·e−4t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (4s + 11)/(s² + 11s + 28)
A =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 5y′ + 6y = 6·u(t), with y(0) = 4 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.1).

y″ + 5y′ + 6y = 6·u(t)y(0) = 4y′(0) = 0u(t) = 10t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 10y′ + 24y = 72·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(0.3).

y″ + 10y′ + 24y = 72·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =

Answer key

  1. 1. f′(0⁺) = −20
  2. 2. f(0⁺) = −7
  3. 3. f(1) = 4.511
  4. 4. A = −1.667
  5. 5. y(0.1) = 4.153
  6. 6. y(0.3) = 12.81

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