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) = (8s² + 23s + 28)/(s(s + 5)(s + 8)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (8s² + 23s + 28)/(s(s + 5)(s + 8))
f(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (3s + 34)/(s² + 9s + 28). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (3s + 34)/(s² + 9s + 28)
f′(0⁺) =
3.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = (3s + 34)/(s² + 10s + 25) and evaluate f(0.3).

F(s) = (3s + 34)/(s² + 10s + 25)t = 0.3 s
f(0.3) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = 37/(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) = 37/(s² + 11s + 28)
A =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 11y′ + 30y = 0 with y(0) = 3 and y′(0) = −6 using the Laplace transform, and find y(0.2).

y″ + 11y′ + 30y = 0y(0) = 3y′(0) = −6t = 0.2 s
y(0.2) =
6.Hard
ReIm−1−2−312

A system is described by the differential equation y″ + 6y′ + 13y = 2u′ + 2u, 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 = 0.2 s. Find g(0.2).

y″ + 6y′ + 13y = 2u′ + 2ut = 0.2 s
g(0.2) =

Answer key

  1. 1. f(0⁺) = 8
  2. 2. f′(0⁺) = 7
  3. 3. f(0.3) = 1.941
  4. 4. A = 12.33
  5. 5. y(0.2) = 1.704
  6. 6. g(0.2) = 0.5835

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