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(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−8s + 27)/(s² + 12s + 35). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−8s + 27)/(s² + 12s + 35)
f(0⁺) =
2.Easy
t (s)f(t)t = 1f(1) = ?

Find the inverse Laplace transform f(t) of F(s) = 12s/(s² + 36) with a table of transform pairs, and evaluate f(1).

F(s) = 12s/(s² + 36)t = 1 s
f(1) =
3.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (4s + 36)/(s² + 10s + 25) and evaluate f(0.25).

F(s) = (4s + 36)/(s² + 10s + 25)t = 0.25 s
f(0.25) =
4.Medium
ReIm−2−4−6−8−10−12−14−162

Find f(t) for F(s) = (s + 14)/(s² + 8s + 25) by completing the square, and evaluate f(0.3).

F(s) = (s + 14)/(s² + 8s + 25)t = 0.3 s
f(0.3) =
5.Hard
ReIm−1−2−3−4−51

A system is described by the differential equation y″ + 6y′ + 5y = 2u′ + 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 = 0.6 s. Find g(0.6).

y″ + 6y′ + 5y = 2u′ + 5ut = 0.6 s
g(0.6) =
6.Hard
ReIm−22

A system is described by the differential equation y″ + 4y′ + 13y = 2u′ + 4u, 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 = 0.1 s. Find g(0.1).

y″ + 4y′ + 13y = 2u′ + 4ut = 0.1 s
g(0.1) =

Answer key

  1. 1. f(0⁺) = −8
  2. 2. f(1) = 11.52
  3. 3. f(0.25) = 2.292
  4. 4. f(0.3) = 0.9737
  5. 5. g(0.6) = 0.4738
  6. 6. g(0.1) = 1.564

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