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) = 8(s + 2)/(s(s + 3)(s + 4)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 8(s + 2)/(s(s + 3)(s + 4))
f(∞) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−s² + 27s + 51)/(s(s + 3)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−s² + 27s + 51)/(s(s + 3)(s + 7))
f(0⁺) =
3.Medium
ReIm−1−2−3−4−5−61

F(s) = (3s + 2)/(s² + 9s + 18). Its inverse transform has the form f(t) = A·e−3t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (3s + 2)/(s² + 9s + 18)
A =
4.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = 10/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.25).

F(s) = 10/(s(s + 4)(s + 5))t = 0.25 s
f(0.25) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

A system obeys y″ + 5y′ + 4y = 8·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(0.2).

y″ + 5y′ + 4y = 8·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.2 s
y(0.2) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

A system obeys y″ + 6y′ + 5y = 10·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 10 for t ≥ 0. Use the Laplace transform to find the output y(0.6).

y″ + 6y′ + 5y = 10·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.6 s
y(0.6) =

Answer key

  1. 1. f(∞) = 1.333
  2. 2. f(0⁺) = −1
  3. 3. A = −2.333
  4. 4. f(0.25) = 0.1533
  5. 5. y(0.2) = 0.5814
  6. 6. y(0.6) = 6.529

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