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) = (s² + 3s + 27)/(s(s + 2)(s + 5)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (s² + 3s + 27)/(s(s + 2)(s + 5))
f(0⁺) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (8s + 11)/(s² + 8s + 18). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (8s + 11)/(s² + 8s + 18)
f′(0⁺) =
3.Medium
ReIm−1−2−3−4−5−61

F(s) = (6s + 20)/(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 + 20)/(s² + 11s + 30)
A =
4.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 10/(s(s + 5)(s + 6)) by partial fractions, and evaluate f(0.3).

F(s) = 10/(s(s + 5)(s + 6))t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 7y′ + 12y = 36·u(t), with y(0) = 4 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.2).

y″ + 7y′ + 12y = 36·u(t)y(0) = 4y′(0) = 0u(t) = 1t = 0.2 s
y(0.2) =
6.Hard
ReIm−1−2−3−4−5−61

A system is described by the differential equation y″ + 6y′ + 8y = 3u′ + 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 = 0.2 s. Find g(0.2).

y″ + 6y′ + 8y = 3u′ + 18ut = 0.2 s
g(0.2) =

Answer key

  1. 1. f(0⁺) = 1
  2. 2. f′(0⁺) = −53
  3. 3. A = −10
  4. 4. f(0.3) = 0.1626
  5. 5. y(0.2) = 3.847
  6. 6. g(0.2) = 2.674

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