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 + 18)/(s² + 8s + 12). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (−8s + 18)/(s² + 8s + 12)
f(0⁺) =
2.Easy
t (s)f(t)f(∞) = ?

A signal f(t) has the Laplace transform F(s) = 14/(s(s² + 7s + 25)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 14/(s(s² + 7s + 25))
f(∞) =
3.Medium
ReIm−1−2−3−4−5−61

F(s) = 38/(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) = 38/(s² + 7s + 6)
A =
4.Medium
ReIm−1−2−3−4−5−61

F(s) = 2/(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) = 2/(s² + 7s + 6)
A =
5.Hard
ReIm−5−10−15

A system is described by the differential equation y″ + 9y′ + 20y = u′ + 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 = 0.1 s. Find g(0.1).

y″ + 9y′ + 20y = u′ + 17ut = 0.1 s
g(0.1) =
6.Hard
t (s)y(t)t = 0.3y(0.3) = ?

A system obeys y″ + 9y′ + 20y = 40·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 5 for t ≥ 0. Use the Laplace transform to find the output y(0.3).

y″ + 9y′ + 20y = 40·u(t)u(t) = 5y(0) = 0, y′(0) = 0t = 0.3 s
y(0.3) =

Answer key

  1. 1. f(0⁺) = −8
  2. 2. f(∞) = 0.56
  3. 3. A = 7.6
  4. 4. A = 0.4
  5. 5. g(0.1) = 1.436
  6. 6. y(0.3) = 3.865

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