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) = 18/(s(s² + 4s + 22)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 18/(s(s² + 4s + 22))
f(∞) =
2.Easy
t (s)f(t)t = 0.2f(0.2) = ?

Find the inverse Laplace transform f(t) of F(s) = 4/(s + 6) − 9/s with a table of transform pairs, and evaluate f(0.2).

F(s) = 4/(s + 6) − 9/st = 0.2 s
f(0.2) =
3.Medium
ReIm−0.5−1−1.5−2−2.5−30.51

Find f(t) for F(s) = 25/(s² + 6s + 10) by completing the square, and evaluate f(0.25).

F(s) = 25/(s² + 6s + 10)t = 0.25 s
f(0.25) =
4.Medium
ReIm−1−2−3−4−5−6−7−81

F(s) = 31/(s² + 10s + 21). Its inverse transform has the form f(t) = A·e−3t + B·e−7t. Find the coefficient A by partial fractions.

F(s) = 31/(s² + 10s + 21)
A =
5.Hard
t (s)y(t)t = 1.2y(1.2) = ?

A system obeys y″ + 4y′ + 3y = 9·u(t), with y(0) = 3 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(1.2).

y″ + 4y′ + 3y = 9·u(t)y(0) = 3y′(0) = 0u(t) = 10t = 1.2 s
y(1.2) =
6.Hard
t (s)y(t)t = 0.75y(0.75) = ?

A system obeys y″ + 6y′ + 5y = 15·u(t), with y(0) = 5 and y′(0) = 0. At t = 0 a step input u(t) = 10 is applied. Use the Laplace transform to find y(0.75).

y″ + 6y′ + 5y = 15·u(t)y(0) = 5y′(0) = 0u(t) = 10t = 0.75 s
y(0.75) =

Answer key

  1. 1. f(∞) = 0.8182
  2. 2. f(0.2) = −7.795
  3. 3. f(0.25) = 2.922
  4. 4. A = 7.75
  5. 5. y(1.2) = 18.17
  6. 6. y(0.75) = 15.39

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