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(∞) = ?

A signal f(t) has the Laplace transform F(s) = 4(s + 4)/(s(s + 1)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 4(s + 4)/(s(s + 1)(s + 8))
f(∞) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−s + 29)/(s² + 9s + 3). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−s + 29)/(s² + 9s + 3)
f′(0⁺) =
3.Medium
ReIm−1−2−3−412

Find f(t) for F(s) = (3s + 9)/(s² + 8s + 20) by completing the square, and evaluate f(0.4).

F(s) = (3s + 9)/(s² + 8s + 20)t = 0.4 s
f(0.4) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = (3s + 22)/(s² + 9s + 14). Its inverse transform has the form f(t) = A·e−2t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (3s + 22)/(s² + 9s + 14)
A =
5.Hard
ReIm−1−2−3−412

A system is described by the differential equation y″ + 6y′ + 13y = 4u′ + 17u, 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 = 0.2 s. Find g(0.2).

y″ + 6y′ + 13y = 4u′ + 17ut = 0.2 s
g(0.2) =
6.Hard
t (s)y(t)t = 0.5y(0.5) = ?

A system obeys y″ + 6y′ + 8y = 8·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(0.5).

y″ + 6y′ + 8y = 8·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.5 s
y(0.5) =

Answer key

  1. 1. f(∞) = 2
  2. 2. f′(0⁺) = 38
  3. 3. f(0.4) = 0.2047
  4. 4. A = 3.2
  5. 5. g(0.2) = 2.556
  6. 6. y(0.5) = 0.7992

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