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(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (7s + 25)/(s² + 7s + 6). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (7s + 25)/(s² + 7s + 6)
f(0⁺) =
2.Easy
t (s)f(t)t = 0.3f(0.3) = ?

Find the inverse Laplace transform f(t) of F(s) = 2/(s + 2)² with a table of transform pairs, and evaluate f(0.3).

F(s) = 2/(s + 2)²t = 0.3 s
f(0.3) =
3.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = (5s + 38)/(s² + 6s + 9) and evaluate f(0.2).

F(s) = (5s + 38)/(s² + 6s + 9)t = 0.2 s
f(0.2) =
4.Medium
ReIm−1−2−3−4−5−6−7−8−91

F(s) = 39/(s² + 10s + 16). Its inverse transform has the form f(t) = A·e−2t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = 39/(s² + 10s + 16)
A =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 2 and y′(0) = −2 using the Laplace transform, and find y(0.25).

y″ + 6y′ + 8y = 0y(0) = 2y′(0) = −2t = 0.25 s
y(0.25) =
6.Hard
t (s)y(t)t = 0.6y(0.6) = ?

A system obeys y″ + 8y′ + 12y = 36·u(t), with y(0) = 2 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.6).

y″ + 8y′ + 12y = 36·u(t)y(0) = 2y′(0) = 0u(t) = 4t = 0.6 s
y(0.6) =

Answer key

  1. 1. f(0⁺) = 7
  2. 2. f(0.3) = 0.3293
  3. 3. f(0.2) = 5.269
  4. 4. A = 6.5
  5. 5. y(0.25) = 1.452
  6. 6. y(0.6) = 7.619

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