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)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 5/(s² + 9) with a table of transform pairs, and evaluate f(0.1).

F(s) = 5/(s² + 9)t = 0.1 s
f(0.1) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 33/(s(s² + 6s + 26)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 33/(s(s² + 6s + 26))
f(∞) =
3.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = (4s + 7)/(s² + 6s + 9) and evaluate f(0.1).

F(s) = (4s + 7)/(s² + 6s + 9)t = 0.1 s
f(0.1) =
4.Medium
t (s)f(t)t = 0.6f(0.6) = ?

Find f(t) for F(s) = 35/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(0.6).

F(s) = 35/(s(s + 1)(s + 4))t = 0.6 s
f(0.6) =
5.Hard
t (s)y(t)t = 1y(1) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = −9 and y′(0) = −12 using the Laplace transform, and find y(1).

y″ + 3y′ + 2y = 0y(0) = −9y′(0) = −12t = 1 s
y(1) =
6.Hard
ReIm−2−4−6−8−102

A system is described by the differential equation y″ + 2y′ + 10y = 2u′ + 18u, 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″ + 2y′ + 10y = 2u′ + 18ut = 0.75 s
g(0.75) =

Answer key

  1. 1. f(0.1) = 0.4925
  2. 2. f(∞) = 1.269
  3. 3. f(0.1) = 2.593
  4. 4. f(0.6) = 2.612
  5. 5. y(1) = −8.194
  6. 6. g(0.75) = 1.367

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