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) = 8(s + 4)/(s(s + 2)(s + 3)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 8(s + 4)/(s(s + 2)(s + 3))
f(∞) =
2.Easy
t (s)f(t)t = 0.2f(0.2) = ?

Find the inverse Laplace transform f(t) of F(s) = 11/(s² + 9) with a table of transform pairs, and evaluate f(0.2).

F(s) = 11/(s² + 9)t = 0.2 s
f(0.2) =
3.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = 20/(s(s + 2)(s + 5)) by partial fractions, and evaluate f(0.5).

F(s) = 20/(s(s + 2)(s + 5))t = 0.5 s
f(0.5) =
4.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 20/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.3).

F(s) = 20/(s(s + 4)(s + 5))t = 0.3 s
f(0.3) =
5.Hard
ReIm−2−4−6−8−10

A system is described by the differential equation y″ + 8y′ + 12y = 2u′ + 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 = 0.75 s. Find g(0.75).

y″ + 8y′ + 12y = 2u′ + 20ut = 0.75 s
g(0.75) =
6.Hard
ReIm−1−212

A system is described by the differential equation y″ + 4y′ + 8y = 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 = 0.6 s. Find g(0.6).

y″ + 4y′ + 8y = ut = 0.6 s
g(0.6) =

Answer key

  1. 1. f(∞) = 5.333
  2. 2. f(0.2) = 2.07
  3. 3. f(0.5) = 0.8832
  4. 4. f(0.3) = 0.3865
  5. 5. g(0.75) = 0.8703
  6. 6. g(0.6) = 0.1404

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