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) = (8s + 19)/(s² + 4s + 24). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (8s + 19)/(s² + 4s + 24)
f′(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−2s + 35)/(s² + 3s + 5). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−2s + 35)/(s² + 3s + 5)
f′(0⁺) =
3.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = 60/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.1).

F(s) = 60/(s(s + 4)(s + 5))t = 0.1 s
f(0.1) =
4.Medium
ReIm−2−4−6−8−10

F(s) = (4s + 35)/(s² + 15s + 56). Its inverse transform has the form f(t) = A·e−7t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = (4s + 35)/(s² + 15s + 56)
A =
5.Hard
ReIm−2−424

A system is described by the differential equation y″ + 2y′ + 17y = u′ + 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 = 1.2 s. Find g(1.2).

y″ + 2y′ + 17y = u′ + 4ut = 1.2 s
g(1.2) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 8y′ + 12y = 12·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.25).

y″ + 8y′ + 12y = 12·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.25 s
y(0.25) =

Answer key

  1. 1. f′(0⁺) = −13
  2. 2. f′(0⁺) = 41
  3. 3. f(0.1) = 0.2236
  4. 4. A = 7
  5. 5. g(1.2) = −0.1987
  6. 6. y(0.25) = 0.2018

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