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)t = 1f(1) = ?

Find the inverse Laplace transform f(t) of F(s) = 12s/(s² + 25) with a table of transform pairs, and evaluate f(1).

F(s) = 12s/(s² + 25)t = 1 s
f(1) =
2.Easy
t (s)f(t)t = 0.75f(0.75) = ?

Find the inverse Laplace transform f(t) of F(s) = 10s/(s² + 25) with a table of transform pairs, and evaluate f(0.75).

F(s) = 10s/(s² + 25)t = 0.75 s
f(0.75) =
3.Medium
t (s)f(t)t = 0.5f(0.5) = ?

Find f(t) for F(s) = 25/(s(s + 3)(s + 5)) by partial fractions, and evaluate f(0.5).

F(s) = 25/(s(s + 3)(s + 5))t = 0.5 s
f(0.5) =
4.Medium
ReIm−1−2−3−4−51

F(s) = (5s + 26)/(s² + 7s + 12). Its inverse transform has the form f(t) = A·e−3t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = (5s + 26)/(s² + 7s + 12)
A =
5.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 5y′ + 4y = 4·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 5 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 5y′ + 4y = 4·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =
6.Hard
ReIm−2−4−6−8−10−12−14−16

A system is described by the differential equation y″ + 6y′ + 10y = u′ + 14u, with input u(t) and output y(t). Find its transfer function G(s) = Y(s)/U(s) (zero initial conditions) and evaluate the impulse response g(t) at t = 0.25 s. Find g(0.25).

y″ + 6y′ + 10y = u′ + 14ut = 0.25 s
g(0.25) =

Answer key

  1. 1. f(1) = 3.404
  2. 2. f(0.75) = −8.206
  3. 3. f(0.5) = 0.9422
  4. 4. A = 11
  5. 5. y(0.3) = 0.5632
  6. 6. g(0.25) = 1.743

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