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) = 2(s + 2)/(s(s + 1)(s + 5)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 2(s + 2)/(s(s + 1)(s + 5))
f(∞) =
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
ReIm−1−2−3−4−5−61

F(s) = (2s + 3)/(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) = (2s + 3)/(s² + 9s + 18)
A =
4.Medium
ReIm2

Find f(t) for F(s) = 7/(s² + 2s + 10) by completing the square, and evaluate f(0.5).

F(s) = 7/(s² + 2s + 10)t = 0.5 s
f(0.5) =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 6y′ + 5y = 7.5·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″ + 6y′ + 5y = 7.5·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =
6.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 8y′ + 15y = 30·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(0.1).

y″ + 8y′ + 15y = 30·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.1 s
y(0.1) =

Answer key

  1. 1. f(∞) = 0.8
  2. 2. f(0.25) = 2.637
  3. 3. A = −1
  4. 4. f(0.5) = 1.412
  5. 5. y(0.3) = 0.9732
  6. 6. y(0.1) = 0.5775

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