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) = (3s² + 13s + 16)/(s(s + 5)(s + 6)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (3s² + 13s + 16)/(s(s + 5)(s + 6))
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 20(s + 6)/(s(s + 5)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 20(s + 6)/(s(s + 5)(s + 8))
f(∞) =
3.Medium
ReIm−2−4−62

Find f(t) for F(s) = (3s + 19)/(s² + 4s + 13) by completing the square, and evaluate f(0.3).

F(s) = (3s + 19)/(s² + 4s + 13)t = 0.3 s
f(0.3) =
4.Medium
ReIm−0.5−1−1.5−2−2.5−30.51

F(s) = 18/(s² + 5s + 6). Its inverse transform has the form f(t) = A·e−2t + B·e−3t. Find the coefficient A by partial fractions.

F(s) = 18/(s² + 5s + 6)
A =
5.Hard
t (s)y(t)t = 0.5y(0.5) = ?

Solve y″ + 8y′ + 12y = 0 with y(0) = 8 and y′(0) = 3 using the Laplace transform, and find y(0.5).

y″ + 8y′ + 12y = 0y(0) = 8y′(0) = 3t = 0.5 s
y(0.5) =
6.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 4y′ + 3y = 9·u(t), with y(0) = 4 and y′(0) = 0. At t = 0 a step input u(t) = 4 is applied. Use the Laplace transform to find y(0.25).

y″ + 4y′ + 3y = 9·u(t)y(0) = 4y′(0) = 0u(t) = 4t = 0.25 s
y(0.25) =

Answer key

  1. 1. f(0⁺) = 3
  2. 2. f(∞) = 3
  3. 3. f(0.3) = 2.886
  4. 4. A = 18
  5. 5. y(0.5) = 4.454
  6. 6. y(0.25) = 4.544

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