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) = (4s + 5)/(s² + 6s + 2). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (4s + 5)/(s² + 6s + 2)
f′(0⁺) =
2.Easy
t (s)f(t)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 17/(s² + 16) with a table of transform pairs, and evaluate f(0.1).

F(s) = 17/(s² + 16)t = 0.1 s
f(0.1) =
3.Medium
ReIm−0.5−1−1.5−20.51

Find f(t) for F(s) = 21/(s² + 4s + 5) by completing the square, and evaluate f(0.2).

F(s) = 21/(s² + 4s + 5)t = 0.2 s
f(0.2) =
4.Medium
ReIm−2−4−6−8−10

F(s) = (3s + 30)/(s² + 11s + 24). Its inverse transform has the form f(t) = A·e−3t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = (3s + 30)/(s² + 11s + 24)
A =
5.Hard
t (s)y(t)t = 0.75y(0.75) = ?

A system obeys y″ + 3y′ + 2y = 4·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.75).

y″ + 3y′ + 2y = 4·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.75 s
y(0.75) =
6.Hard
ReIm−2−4−6−8−10−12−142

A system is described by the differential equation y″ + 2y′ + 10y = u′ + 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 = 0.3 s. Find g(0.3).

y″ + 2y′ + 10y = u′ + 13ut = 0.3 s
g(0.3) =

Answer key

  1. 1. f′(0⁺) = −19
  2. 2. f(0.1) = 1.655
  3. 3. f(0.2) = 2.797
  4. 4. A = 4.2
  5. 5. y(0.75) = 0.5568
  6. 6. g(0.3) = 2.782

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