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) = (−s + 8)/(s² + 7s + 6). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−s + 8)/(s² + 7s + 6)
f(0⁺) =
2.Easy
t (s)f(t)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 16s/(s² + 9) with a table of transform pairs, and evaluate f(0.25).

F(s) = 16s/(s² + 9)t = 0.25 s
f(0.25) =
3.Medium
ReIm−5−105

Find f(t) for F(s) = (s + 11)/(s² + 4s + 29) by completing the square, and evaluate f(0.1).

F(s) = (s + 11)/(s² + 4s + 29)t = 0.1 s
f(0.1) =
4.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (3s + 35)/(s² + 6s + 9) and evaluate f(0.25).

F(s) = (3s + 35)/(s² + 6s + 9)t = 0.25 s
f(0.25) =
5.Hard
ReIm−0.5−1−1.5−2−2.5−30.51

A system is described by the differential equation y″ + 4y′ + 3y = 4u′ + 9u, 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″ + 4y′ + 3y = 4u′ + 9ut = 0.75 s
g(0.75) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

A system obeys y″ + 6y′ + 8y = 8·u(t), with y(0) = 3 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.6).

y″ + 6y′ + 8y = 8·u(t)y(0) = 3y′(0) = 0u(t) = 4t = 0.6 s
y(0.6) =

Answer key

  1. 1. f(0⁺) = −1
  2. 2. f(0.25) = 11.71
  3. 3. f(0.1) = 1.425
  4. 4. f(0.25) = 4.487
  5. 5. g(0.75) = 1.339
  6. 6. y(0.6) = 3.488

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