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 = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 12/(s + 6) + 4/s with a table of transform pairs, and evaluate f(0.1).

F(s) = 12/(s + 6) + 4/st = 0.1 s
f(0.1) =
2.Easy
t (s)f(t)t = 0.1f(0.1) = ?

Find the inverse Laplace transform f(t) of F(s) = 6s/(s² + 9) with a table of transform pairs, and evaluate f(0.1).

F(s) = 6s/(s² + 9)t = 0.1 s
f(0.1) =
3.Medium
t (s)f(t)t = 0.8f(0.8) = ?

Find f(t) for F(s) = (s + 37)/(s² + 4s + 4) and evaluate f(0.8).

F(s) = (s + 37)/(s² + 4s + 4)t = 0.8 s
f(0.8) =
4.Medium
t (s)f(t)t = 0.3f(0.3) = ?

Find f(t) for F(s) = 60/(s(s + 5)(s + 6)) by partial fractions, and evaluate f(0.3).

F(s) = 60/(s(s + 5)(s + 6))t = 0.3 s
f(0.3) =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 9y′ + 18y = 18·u(t), with y(0) = 1 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″ + 9y′ + 18y = 18·u(t)y(0) = 1y′(0) = 0u(t) = 4t = 0.25 s
y(0.25) =
6.Hard
ReIm−0.5−1−1.5−2−2.5−30.51

A system is described by the differential equation y″ + 5y′ + 6y = 7u, 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 = 0.4 s. Find g(0.4).

y″ + 5y′ + 6y = 7ut = 0.4 s
g(0.4) =

Answer key

  1. 1. f(0.1) = 10.59
  2. 2. f(0.1) = 5.732
  3. 3. f(0.8) = 5.855
  4. 4. f(0.3) = 0.9754
  5. 5. y(0.25) = 1.835
  6. 6. g(0.4) = 1.037

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