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)t = 0.2f(0.2) = ?

Find the inverse Laplace transform f(t) of F(s) = 9/(s + 6) + 6/s with a table of transform pairs, and evaluate f(0.2).

F(s) = 9/(s + 6) + 6/st = 0.2 s
f(0.2) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (5s + 24)/(s² + 4s + 26). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (5s + 24)/(s² + 4s + 26)
f′(0⁺) =
3.Medium
ReIm−2−4−6−8−10

F(s) = (s + 9)/(s² + 12s + 35). Its inverse transform has the form f(t) = A·e−5t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = (s + 9)/(s² + 12s + 35)
A =
4.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 15/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.3).

F(s) = 15/(s(s + 4)(s + 5))t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 3y′ + 2y = 6·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(0.3).

y″ + 3y′ + 2y = 6·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =
6.Hard
ReIm−2−42

A system is described by the differential equation y″ + 6y′ + 18y = 3u′ + 13u, 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 = 0.4 s. Find g(0.4).

y″ + 6y′ + 18y = 3u′ + 13ut = 0.4 s
g(0.4) =

Answer key

  1. 1. f(0.2) = 8.711
  2. 2. f′(0⁺) = 4
  3. 3. A = 2
  4. 4. f(0.3) = 0.2899
  5. 5. y(0.3) = 1.008
  6. 6. g(0.4) = 0.7017

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