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) = (−8s + 9)/(s² + 13s + 40). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−8s + 9)/(s² + 13s + 40)
f(0⁺) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (−9s + 32)/(s² + 12s + 32). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−9s + 32)/(s² + 12s + 32)
f(0⁺) =
3.Medium
t (s)f(t)t = 0.1f(0.1) = ?

Find f(t) for F(s) = 60/(s(s + 4)(s + 5)) by partial fractions, and evaluate f(0.1).

F(s) = 60/(s(s + 4)(s + 5))t = 0.1 s
f(0.1) =
4.Medium
ReIm−1−2−3−4−5−6−71

F(s) = (6s + 40)/(s² + 4s + 3). Its inverse transform has the form f(t) = A·e−t + B·e−3t. Find the coefficient A by partial fractions.

F(s) = (6s + 40)/(s² + 4s + 3)
A =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

A system obeys y″ + 3y′ + 2y = 8·u(t), with y(0) = 3 and y′(0) = 0. At t = 0 a step input u(t) = 1 is applied. Use the Laplace transform to find y(0.25).

y″ + 3y′ + 2y = 8·u(t)y(0) = 3y′(0) = 0u(t) = 1t = 0.25 s
y(0.25) =
6.Hard
ReIm−2−4−6−8−10

A system is described by the differential equation y″ + 6y′ + 8y = u′ + 10u, 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 = 0.6 s. Find g(0.6).

y″ + 6y′ + 8y = u′ + 10ut = 0.6 s
g(0.6) =

Answer key

  1. 1. f(0⁺) = −8
  2. 2. f(0⁺) = −9
  3. 3. f(0.1) = 0.2236
  4. 4. A = 17
  5. 5. y(0.25) = 3.049
  6. 6. g(0.6) = 0.9326

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