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(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−9s + 24)/(s² + 10s + 24). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−9s + 24)/(s² + 10s + 24)
f(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−7s² + 7s + 15)/(s(s + 4)(s + 6)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−7s² + 7s + 15)/(s(s + 4)(s + 6))
f(0⁺) =
3.Medium
ReIm−2−4−6−8−10−12−14

F(s) = (3s + 40)/(s² + 14s + 48). Its inverse transform has the form f(t) = A·e−6t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = (3s + 40)/(s² + 14s + 48)
A =
4.Medium
ReIm−2−4−6−824

Find f(t) for F(s) = (3s + 22)/(s² + 6s + 25) by completing the square, and evaluate f(0.2).

F(s) = (3s + 22)/(s² + 6s + 25)t = 0.2 s
f(0.2) =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

Solve y″ + 9y′ + 18y = 0 with y(0) = 10 and y′(0) = −15 using the Laplace transform, and find y(0.1).

y″ + 9y′ + 18y = 0y(0) = 10y′(0) = −15t = 0.1 s
y(0.1) =
6.Hard
ReIm−1−212

A system is described by the differential equation y″ + 4y′ + 8y = 2u, 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″ + 4y′ + 8y = 2ut = 0.1 s
g(0.1) =

Answer key

  1. 1. f(0⁺) = −9
  2. 2. f(0⁺) = −7
  3. 3. A = 11
  4. 4. f(0.2) = 2.427
  5. 5. y(0.1) = 8.368
  6. 6. g(0.1) = 0.1627

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