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) = (−9s² + 2s + 41)/(s(s + 1)(s + 3)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−9s² + 2s + 41)/(s(s + 1)(s + 3))
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 20(s + 1)/(s(s + 2)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 20(s + 1)/(s(s + 2)(s + 8))
f(∞) =
3.Medium
ReIm−5−10−15−20−25−30

F(s) = (s + 27)/(s² + 7s + 6). Its inverse transform has the form f(t) = A·e−t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (s + 27)/(s² + 7s + 6)
A =
4.Medium
t (s)f(t)t = 0.4f(0.4) = ?

Find f(t) for F(s) = 60/(s(s + 1)(s + 2)) by partial fractions, and evaluate f(0.4).

F(s) = 60/(s(s + 1)(s + 2))t = 0.4 s
f(0.4) =
5.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.50.51

A system is described by the differential equation y″ + 5y′ + 4y = u′ + 2u, 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 = 0.2 s. Find g(0.2).

y″ + 5y′ + 4y = u′ + 2ut = 0.2 s
g(0.2) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 8y′ + 15y = 60·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 10 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 8y′ + 15y = 60·u(t)u(t) = 10y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(0⁺) = −9
  2. 2. f(∞) = 1.25
  3. 3. A = 5.2
  4. 4. f(0.4) = 3.261
  5. 5. g(0.2) = 0.5725
  6. 6. y(0.3) = 12.73

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