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) = (7s + 13)/(s² + 6s + 12). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (7s + 13)/(s² + 6s + 12)
f′(0⁺) =
2.Easy
t (s)f(t)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 11/(s² + 4) with a table of transform pairs, and evaluate f(0.25).

F(s) = 11/(s² + 4)t = 0.25 s
f(0.25) =
3.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (6s + 31)/(s² + 6s + 9) and evaluate f(0.25).

F(s) = (6s + 31)/(s² + 6s + 9)t = 0.25 s
f(0.25) =
4.Medium
ReIm−1−2−3−4−5−61

F(s) = (6s + 4)/(s² + 11s + 30). Its inverse transform has the form f(t) = A·e−5t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (6s + 4)/(s² + 11s + 30)
A =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 7y′ + 12y = 48·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.25).

y″ + 7y′ + 12y = 48·u(t)y(0) = 1y′(0) = 0u(t) = 4t = 0.25 s
y(0.25) =
6.Hard
ReIm−2−4−6−8−10

A system is described by the differential equation y″ + 3y′ + 2y = 2u′ + 18u, 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 = 0.8 s. Find g(0.8).

y″ + 3y′ + 2y = 2u′ + 18ut = 0.8 s
g(0.8) =

Answer key

  1. 1. f′(0⁺) = −29
  2. 2. f(0.25) = 2.637
  3. 3. f(0.25) = 4.369
  4. 4. A = −26
  5. 5. y(0.25) = 4.213
  6. 6. g(0.8) = 4.363

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