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) = 19(s + 6)/(s(s + 2)(s + 5)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 19(s + 6)/(s(s + 2)(s + 5))
f(∞) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 11(s + 4)/(s(s + 2)(s + 3)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 11(s + 4)/(s(s + 2)(s + 3))
f(∞) =
3.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 10/(s(s + 1)(s + 6)) by partial fractions, and evaluate f(0.3).

F(s) = 10/(s(s + 1)(s + 6))t = 0.3 s
f(0.3) =
4.Medium
ReIm−1−2−3−4−5−6−71

F(s) = (6s + 40)/(s² + 6s + 5). Its inverse transform has the form f(t) = A·e−t + B·e−5t. Find the coefficient A by partial fractions.

F(s) = (6s + 40)/(s² + 6s + 5)
A =
5.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 = 0.25 s. Find g(0.25).

y″ + 4y′ + 8y = 2ut = 0.25 s
g(0.25) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 7 and y′(0) = 14 using the Laplace transform, and find y(0.6).

y″ + 4y′ + 3y = 0y(0) = 7y′(0) = 14t = 0.6 s
y(0.6) =

Answer key

  1. 1. f(∞) = 11.4
  2. 2. f(∞) = 7.333
  3. 3. f(0.3) = 0.2401
  4. 4. A = 8.5
  5. 5. g(0.25) = 0.2908
  6. 6. y(0.6) = 7.869

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