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) = 32/(s(s² + 4s + 22)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 32/(s(s² + 4s + 22))
f(∞) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 3/(s(s² + 2s + 24)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 3/(s(s² + 2s + 24))
f(∞) =
3.Medium
ReIm−1−2−3−4−51

F(s) = 25/(s² + 6s + 5). Its inverse transform has the form f(t) = A·e−t + B·e−5t. Find the coefficient A by partial fractions.

F(s) = 25/(s² + 6s + 5)
A =
4.Medium
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

F(s) = 23/(s² + 5s + 4). Its inverse transform has the form f(t) = A·e−t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = 23/(s² + 5s + 4)
A =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = −11 using the Laplace transform, and find y(0.25).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = −11t = 0.25 s
y(0.25) =
6.Hard
t (s)y(t)t = 1.2y(1.2) = ?

A system obeys y″ + 4y′ + 3y = 9·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(1.2).

y″ + 4y′ + 3y = 9·u(t)y(0) = 1y′(0) = 0u(t) = 2t = 1.2 s
y(1.2) =

Answer key

  1. 1. f(∞) = 1.455
  2. 2. f(∞) = 0.125
  3. 3. A = 6.25
  4. 4. A = 7.667
  5. 5. y(0.25) = −0.4674
  6. 6. y(1.2) = 3.809

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