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

A signal f(t) has the Laplace transform F(s) = (6s² + 14s + 14)/(s(s + 2)(s + 7)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (6s² + 14s + 14)/(s(s + 2)(s + 7))

f(0⁺) =

2.Easy

A signal f(t) has the Laplace transform F(s) = 3(s + 6)/(s(s + 4)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 3(s + 6)/(s(s + 4)(s + 8))

f(∞) =

3.Medium

Find f(t) for F(s) = 35/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(0.75).

F(s) = 35/(s(s + 1)(s + 4))t = 0.75 s

f(0.75) =

4.Medium

Find f(t) for F(s) = 25/(s(s + 1)(s + 4)) by partial fractions, and evaluate f(0.75).

F(s) = 25/(s(s + 1)(s + 4))t = 0.75 s

f(0.75) =

5.Hard

A system is described by the differential equation y″ + 4y′ + 20y = 2u′ + 9u, 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 = 0.75 s. Find g(0.75).

y″ + 4y′ + 20y = 2u′ + 9ut = 0.75 s

g(0.75) =

6.Hard

A system obeys y″ + 9y′ + 20y = 120·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.25).

y″ + 9y′ + 20y = 120·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.25 s

y(0.25) =

Answer key

1. f(0⁺) = 6
2. f(∞) = 0.5625
3. f(0.75) = 3.384
4. f(0.75) = 2.417
5. g(0.75) = −0.4024
6. y(0.25) = 1.84

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