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) = 4(s + 1)/(s(s + 6)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 4(s + 1)/(s(s + 6)(s + 7))
f(∞) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−5s + 40)/(s² + 10s + 24). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−5s + 40)/(s² + 10s + 24)
f(0⁺) =
3.Medium
t (s)f(t)t = 0.75f(0.75) = ?

Find f(t) for F(s) = 15/(s(s + 2)(s + 4)) by partial fractions, and evaluate f(0.75).

F(s) = 15/(s(s + 2)(s + 4))t = 0.75 s
f(0.75) =
4.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = (6s + 37)/(s² + 4s + 4) and evaluate f(0.2).

F(s) = (6s + 37)/(s² + 4s + 4)t = 0.2 s
f(0.2) =
5.Hard
t (s)y(t)t = 0.6y(0.6) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = −2 and y′(0) = −13 using the Laplace transform, and find y(0.6).

y″ + 5y′ + 4y = 0y(0) = −2y′(0) = −13t = 0.6 s
y(0.6) =
6.Hard
ReIm−1−2−3−4−5−6−71

A system is described by the differential equation y″ + 7y′ + 6y = 3u′ + 19u, 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 = 1.5 s. Find g(1.5).

y″ + 7y′ + 6y = 3u′ + 19ut = 1.5 s
g(1.5) =

Answer key

  1. 1. f(∞) = 0.09524
  2. 2. f(0⁺) = −5
  3. 3. f(0.75) = 1.132
  4. 4. f(0.2) = 7.374
  5. 5. y(0.6) = −3.388
  6. 6. g(1.5) = 0.714

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