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) = (5s + 24)/(s² + 6s + 30). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (5s + 24)/(s² + 6s + 30)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (9s² + 29s + 22)/(s(s + 4)(s + 8)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (9s² + 29s + 22)/(s(s + 4)(s + 8))
f(0⁺) =
3.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = (5s + 19)/(s² + 10s + 25) and evaluate f(0.1).

F(s) = (5s + 19)/(s² + 10s + 25)t = 0.1 s
f(0.1) =
4.Medium
ReIm−5−10−15−20−25−30−35

F(s) = (s + 33)/(s² + 8s + 12). Its inverse transform has the form f(t) = A·e−2t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (s + 33)/(s² + 8s + 12)
A =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 10y′ + 24y = 0 with y(0) = −10 and y′(0) = −5 using the Laplace transform, and find y(0.2).

y″ + 10y′ + 24y = 0y(0) = −10y′(0) = −5t = 0.2 s
y(0.2) =
6.Hard
ReIm−2−42

A system is described by the differential equation y″ + 2y′ + 10y = 4u′ + 20u, with input u(t) and output y(t). Find its transfer function G(s) = Y(s)/U(s) (zero initial conditions) and evaluate the impulse response g(t) at t = 1 s. Find g(1).

y″ + 2y′ + 10y = 4u′ + 20ut = 1 s
g(1) =

Answer key

  1. 1. f′(0⁺) = −6
  2. 2. f(0⁺) = 9
  3. 3. f(0.1) = 2.669
  4. 4. A = 7.75
  5. 5. y(0.2) = −7.826
  6. 6. g(1) = −1.18

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