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 + 5)/(s² + 8s + 14). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (2s + 5)/(s² + 8s + 14)
f′(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 40/(s(s² + 5s + 3)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 40/(s(s² + 5s + 3))
f(∞) =
3.Medium
t (s)f(t)t = 0.4f(0.4) = ?

Find f(t) for F(s) = (s + 36)/(s² + 6s + 9) and evaluate f(0.4).

F(s) = (s + 36)/(s² + 6s + 9)t = 0.4 s
f(0.4) =
4.Medium
ReIm−1−2−3−4−5−61

F(s) = (3s + 2)/(s² + 7s + 6). Its inverse transform has the form f(t) = A·e−t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (3s + 2)/(s² + 7s + 6)
A =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 8y′ + 12y = 72·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(0.2).

y″ + 8y′ + 12y = 72·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
ReIm−5−10−15−20

A system is described by the differential equation y″ + 2y′ + 10y = u′ + 18u, 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″ + 2y′ + 10y = u′ + 18ut = 0.1 s
g(0.1) =

Answer key

  1. 1. f′(0⁺) = −11
  2. 2. f(∞) = 13.33
  3. 3. f(0.4) = 4.277
  4. 4. A = −0.2
  5. 5. y(0.2) = 1.741
  6. 6. g(0.1) = 2.38

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