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) = (−s + 10)/(s² + 7s + 30). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−s + 10)/(s² + 7s + 30)
f′(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (8s + 18)/(s² + 12s + 32). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (8s + 18)/(s² + 12s + 32)
f(0⁺) =
3.Medium
t (s)f(t)t = 0.8f(0.8) = ?

Find f(t) for F(s) = 35/(s(s + 1)(s + 6)) by partial fractions, and evaluate f(0.8).

F(s) = 35/(s(s + 1)(s + 6))t = 0.8 s
f(0.8) =
4.Medium
ReIm−112

Find f(t) for F(s) = (4s + 2)/(s² + 2s + 5) by completing the square, and evaluate f(1.2).

F(s) = (4s + 2)/(s² + 2s + 5)t = 1.2 s
f(1.2) =
5.Hard
ReIm−0.5−1−1.5−2−2.50.51

A system is described by the differential equation y″ + 4y′ + 5y = 4u′ + 10u, 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 = 0.75 s. Find g(0.75).

y″ + 4y′ + 5y = 4u′ + 10ut = 0.75 s
g(0.75) =
6.Hard
t (s)y(t)t = 0.4y(0.4) = ?

A system obeys y″ + 6y′ + 8y = 24·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 5 for t ≥ 0. Use the Laplace transform to find the output y(0.4).

y″ + 6y′ + 8y = 24·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.4 s
y(0.4) =

Answer key

  1. 1. f′(0⁺) = 17
  2. 2. f(0⁺) = 8
  3. 3. f(0.8) = 2.698
  4. 4. f(1.2) = −1.092
  5. 5. g(0.75) = 0.9572
  6. 6. y(0.4) = 4.549

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