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)

A signal f(t) has the Laplace transform F(s) = (2s + 28)/(s² + 2s + 4). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (2s + 28)/(s² + 2s + 4)
f′(0⁺) =
2.Easy
t (s)f(t)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 15/(s² + 36) with a table of transform pairs, and evaluate f(0.1).

F(s) = 15/(s² + 36)t = 0.1 s
f(0.1) =
3.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = 55/(s(s + 3)(s + 4)) by partial fractions, and evaluate f(0.5).

F(s) = 55/(s(s + 3)(s + 4))t = 0.5 s
f(0.5) =
4.Medium
ReIm−1−2−3−4−5−6−7−8−912

Find f(t) for F(s) = (3s + 25)/(s² + 4s + 8) by completing the square, and evaluate f(0.25).

F(s) = (3s + 25)/(s² + 4s + 8)t = 0.25 s
f(0.25) =
5.Hard
ReIm−1−2−3−4−5−61

A system is described by the differential equation y″ + 8y′ + 12y = 4u′ + 12u, 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 = 0.25 s. Find g(0.25).

y″ + 8y′ + 12y = 4u′ + 12ut = 0.25 s
g(0.25) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

A system obeys y″ + 5y′ + 4y = 12·u(t), with y(0) = 6 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.6).

y″ + 5y′ + 4y = 12·u(t)y(0) = 6y′(0) = 0u(t) = 2t = 0.6 s
y(0.6) =

Answer key

  1. 1. f′(0⁺) = 24
  2. 2. f(0.1) = 1.412
  3. 3. f(0.5) = 2.353
  4. 4. f(0.25) = 4.359
  5. 5. g(0.25) = 1.276
  6. 6. y(0.6) = 6

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