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(∞) = ?

A signal f(t) has the Laplace transform F(s) = 15(s + 4)/(s(s + 3)(s + 5)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 15(s + 4)/(s(s + 3)(s + 5))
f(∞) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (3s + 26)/(s² + 5s + 28). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (3s + 26)/(s² + 5s + 28)
f′(0⁺) =
3.Medium
ReIm24

Find f(t) for F(s) = 18/(s² + 2s + 17) by completing the square, and evaluate f(0.25).

F(s) = 18/(s² + 2s + 17)t = 0.25 s
f(0.25) =
4.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = (6s + 9)/(s² + 8s + 16) and evaluate f(0.1).

F(s) = (6s + 9)/(s² + 8s + 16)t = 0.1 s
f(0.1) =
5.Hard
ReIm−22

A system is described by the differential equation y″ + 4y′ + 13y = 13u, 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 = 0.25 s. Find g(0.25).

y″ + 4y′ + 13y = 13ut = 0.25 s
g(0.25) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 8y′ + 15y = 60·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.3).

y″ + 8y′ + 15y = 60·u(t)y(0) = 6y′(0) = 0u(t) = 10t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(∞) = 4
  2. 2. f′(0⁺) = 11
  3. 3. f(0.25) = 2.949
  4. 4. f(0.1) = 3.016
  5. 5. g(0.25) = 1.792
  6. 6. y(0.3) = 16.82

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