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) = (−2s + 11)/(s² + 7s + 10). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−2s + 11)/(s² + 7s + 10)
f(0⁺) =
2.Easy
t (s)f(t)t = 0.4f(0.4) = ?

Find the inverse Laplace transform f(t) of F(s) = 2/(s² + 25) with a table of transform pairs, and evaluate f(0.4).

F(s) = 2/(s² + 25)t = 0.4 s
f(0.4) =
3.Medium
ReIm−1−2−3−4−5−61

F(s) = (2s + 11)/(s² + 8s + 15). Its inverse transform has the form f(t) = A·e−3t + B·e−5t. Find the coefficient A by partial fractions.

F(s) = (2s + 11)/(s² + 8s + 15)
A =
4.Medium
ReIm−1−2−3−4−5−6−7−8−91

F(s) = (3s + 6)/(s² + 11s + 24). Its inverse transform has the form f(t) = A·e−3t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = (3s + 6)/(s² + 11s + 24)
A =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 10y′ + 24y = 72·u(t), with y(0) = 5 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.1).

y″ + 10y′ + 24y = 72·u(t)y(0) = 5y′(0) = 0u(t) = 1t = 0.1 s
y(0.1) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 8y′ + 12y = 12·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 2 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 8y′ + 12y = 12·u(t)u(t) = 2y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(0⁺) = −2
  2. 2. f(0.4) = 0.3637
  3. 3. A = 2.5
  4. 4. A = −0.6
  5. 5. y(0.1) = 4.827
  6. 6. y(0.3) = 0.5189

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