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) = (−7s + 19)/(s² + 9s + 18). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−7s + 19)/(s² + 9s + 18)
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 19(s + 1)/(s(s + 4)(s + 7)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 19(s + 1)/(s(s + 4)(s + 7))
f(∞) =
3.Medium
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

F(s) = (3s + 11)/(s² + 6s + 8). Its inverse transform has the form f(t) = A·e−2t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = (3s + 11)/(s² + 6s + 8)
A =
4.Medium
t (s)f(t)t = 0.6f(0.6) = ?

Find f(t) for F(s) = (5s + 34)/(s² + 4s + 4) and evaluate f(0.6).

F(s) = (5s + 34)/(s² + 4s + 4)t = 0.6 s
f(0.6) =
5.Hard
t (s)y(t)t = 0.1y(0.1) = ?

A system obeys y″ + 9y′ + 20y = 80·u(t), with y(0) = 1 and y′(0) = 0. At t = 0 a step input u(t) = 2 is applied. Use the Laplace transform to find y(0.1).

y″ + 9y′ + 20y = 80·u(t)y(0) = 1y′(0) = 0u(t) = 2t = 0.1 s
y(0.1) =
6.Hard
ReIm−1−2−3−41

A system is described by the differential equation y″ + 3y′ + 2y = 4u′ + 17u, 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″ + 3y′ + 2y = 4u′ + 17ut = 0.25 s
g(0.25) =

Answer key

  1. 1. f(0⁺) = −7
  2. 2. f(∞) = 0.6786
  3. 3. A = 2.5
  4. 4. f(0.6) = 5.843
  5. 5. y(0.1) = 1.522
  6. 6. g(0.25) = 4.666

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