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) = 3(s + 3)/(s(s + 5)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 3(s + 3)/(s(s + 5)(s + 7))
f(∞) =
2.Easy
t (s)f(t)t = 0.3f(0.3) = ?

Find the inverse Laplace transform f(t) of F(s) = 5/(s² + 36) with a table of transform pairs, and evaluate f(0.3).

F(s) = 5/(s² + 36)t = 0.3 s
f(0.3) =
3.Medium
ReIm−1−2−3−4−5−61

Find f(t) for F(s) = (4s + 23)/(s² + 6s + 10) by completing the square, and evaluate f(0.6).

F(s) = (4s + 23)/(s² + 6s + 10)t = 0.6 s
f(0.6) =
4.Medium
ReIm−1−2−3−4−51

F(s) = 20/(s² + 7s + 10). Its inverse transform has the form f(t) = A·e−2t + B·e−5t. Find the coefficient A by partial fractions.

F(s) = 20/(s² + 7s + 10)
A =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

Solve y″ + 9y′ + 18y = 0 with y(0) = −9 and y′(0) = −3 using the Laplace transform, and find y(0.3).

y″ + 9y′ + 18y = 0y(0) = −9y′(0) = −3t = 0.3 s
y(0.3) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 7y′ + 12y = 24·u(t), with y(0) = 5 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.25).

y″ + 7y′ + 12y = 24·u(t)y(0) = 5y′(0) = 0u(t) = 2t = 0.25 s
y(0.25) =

Answer key

  1. 1. f(∞) = 0.2571
  2. 2. f(0.3) = 0.8115
  3. 3. f(0.6) = 1.572
  4. 4. A = 6.667
  5. 5. y(0.3) = −6.072
  6. 6. y(0.25) = 4.786

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