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 + 23)/(s² + 5s + 14). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−5s + 23)/(s² + 5s + 14)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (5s² + 24s + 53)/(s(s + 3)(s + 6)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (5s² + 24s + 53)/(s(s + 3)(s + 6))
f(0⁺) =
3.Medium
ReIm−2−4−6−82

Find f(t) for F(s) = (s + 7)/(s² + 2s + 10) by completing the square, and evaluate f(0.2).

F(s) = (s + 7)/(s² + 2s + 10)t = 0.2 s
f(0.2) =
4.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = (3s + 20)/(s² + 8s + 16) and evaluate f(0.5).

F(s) = (3s + 20)/(s² + 8s + 16)t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 5y′ + 6y = 6·u(t), with y(0) = 3 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.1).

y″ + 5y′ + 6y = 6·u(t)y(0) = 3y′(0) = 0u(t) = 4t = 0.1 s
y(0.1) =
6.Hard
ReIm−224

A system is described by the differential equation y″ + 4y′ + 20y = 3u′ + u, 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 = 0.1 s. Find g(0.1).

y″ + 4y′ + 20y = 3u′ + ut = 0.1 s
g(0.1) =

Answer key

  1. 1. f′(0⁺) = 48
  2. 2. f(0⁺) = 5
  3. 3. f(0.2) = 1.6
  4. 4. f(0.5) = 0.9473
  5. 5. y(0.1) = 3.025
  6. 6. g(0.1) = 1.864

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