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) = (7s² + 29s + 57)/(s(s + 2)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (7s² + 29s + 57)/(s(s + 2)(s + 7))
f(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (6s + 15)/(s² + 8s + 9). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (6s + 15)/(s² + 8s + 9)
f′(0⁺) =
3.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = (5s + 17)/(s² + 2s + 1) and evaluate f(0.3).

F(s) = (5s + 17)/(s² + 2s + 1)t = 0.3 s
f(0.3) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = 8/(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) = 8/(s² + 10s + 21)
A =
5.Hard
t (s)y(t)t = 1.2y(1.2) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 5 and y′(0) = 11 using the Laplace transform, and find y(1.2).

y″ + 4y′ + 3y = 0y(0) = 5y′(0) = 11t = 1.2 s
y(1.2) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

Solve y″ + 10y′ + 24y = 0 with y(0) = 4 and y′(0) = −10 using the Laplace transform, and find y(0.3).

y″ + 10y′ + 24y = 0y(0) = 4y′(0) = −10t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(0⁺) = 7
  2. 2. f′(0⁺) = −33
  3. 3. f(0.3) = 6.371
  4. 4. A = 2
  5. 5. y(1.2) = 3.697
  6. 6. y(0.3) = 1.612

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