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 = 1.2f(1.2) = ?

Find the inverse Laplace transform f(t) of F(s) = 20/(s² + 9) with a table of transform pairs, and evaluate f(1.2).

F(s) = 20/(s² + 9)t = 1.2 s
f(1.2) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−3s² + 30s + 50)/(s(s + 2)(s + 3)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−3s² + 30s + 50)/(s(s + 2)(s + 3))
f(0⁺) =
3.Medium
ReIm5

Find f(t) for F(s) = 15/(s² + 4s + 29) by completing the square, and evaluate f(0.75).

F(s) = 15/(s² + 4s + 29)t = 0.75 s
f(0.75) =
4.Medium
t (s)f(t)t = 0.8f(0.8) = ?

Find f(t) for F(s) = (5s + 9)/(s² + 4s + 4) and evaluate f(0.8).

F(s) = (5s + 9)/(s² + 4s + 4)t = 0.8 s
f(0.8) =
5.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

A system is described by the differential equation y″ + 6y′ + 8y = 3u′ + 5u, 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 = 0.2 s. Find g(0.2).

y″ + 6y′ + 8y = 3u′ + 5ut = 0.2 s
g(0.2) =
6.Hard
ReIm−1−2−3−4−512

A system is described by the differential equation y″ + 2y′ + 5y = 3u′ + 14u, 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′ + 5y = 3u′ + 14ut = 0.75 s
g(0.75) =

Answer key

  1. 1. f(1.2) = −2.95
  2. 2. f(0⁺) = −3
  3. 3. f(0.75) = −0.3826
  4. 4. f(0.8) = 0.848
  5. 5. g(0.2) = 1.237
  6. 6. g(0.75) = 2.692

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