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) = (−7s + 33)/(s² + 4s + 9). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−7s + 33)/(s² + 4s + 9)
f′(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−5s + 5)/(s² + 6s + 24). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−5s + 5)/(s² + 6s + 24)
f′(0⁺) =
3.Medium
ReIm−0.5−1−1.5−20.51

Find f(t) for F(s) = (5s + 6)/(s² + 4s + 5) by completing the square, and evaluate f(0.5).

F(s) = (5s + 6)/(s² + 4s + 5)t = 0.5 s
f(0.5) =
4.Medium
ReIm−1−2−3−4−5−6−7−8−91

F(s) = (5s + 37)/(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) = (5s + 37)/(s² + 15s + 56)
A =
5.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

A system is described by the differential equation y″ + 5y′ + 4y = 16u, 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 = 0.4 s. Find g(0.4).

y″ + 5y′ + 4y = 16ut = 0.4 s
g(0.4) =
6.Hard
t (s)y(t)t = 1y(1) = ?

A system obeys y″ + 3y′ + 2y = 10·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(1).

y″ + 3y′ + 2y = 10·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 1 s
y(1) =

Answer key

  1. 1. f′(0⁺) = 61
  2. 2. f′(0⁺) = 35
  3. 3. f(0.5) = 0.9087
  4. 4. A = 2
  5. 5. g(0.4) = 2.498
  6. 6. y(1) = 3.996

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