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) = (3s + 34)/(s² + 6s + 11). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (3s + 34)/(s² + 6s + 11)
f′(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (5s + 34)/(s² + 6s + 20). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (5s + 34)/(s² + 6s + 20)
f′(0⁺) =
3.Medium
t (s)f(t)t = 1f(1) = ?

Find f(t) for F(s) = (3s + 37)/(s² + 4s + 4) and evaluate f(1).

F(s) = (3s + 37)/(s² + 4s + 4)t = 1 s
f(1) =
4.Medium
ReIm−5−10−15−20−25−30−35−40

F(s) = (s + 37)/(s² + 6s + 8). Its inverse transform has the form f(t) = A·e−2t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = (s + 37)/(s² + 6s + 8)
A =
5.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.51

A system is described by the differential equation y″ + 6y′ + 10y = 4u′ + 15u, 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″ + 6y′ + 10y = 4u′ + 15ut = 0.1 s
g(0.1) =
6.Hard
t (s)y(t)t = 0.4y(0.4) = ?

A system obeys y″ + 3y′ + 2y = 2·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 2 for t ≥ 0. Use the Laplace transform to find the output y(0.4).

y″ + 3y′ + 2y = 2·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.4 s
y(0.4) =

Answer key

  1. 1. f′(0⁺) = 16
  2. 2. f′(0⁺) = 4
  3. 3. f(1) = 4.601
  4. 4. A = 17.5
  5. 5. g(0.1) = 3.17
  6. 6. y(0.4) = 0.2174

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