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)t = 0.25f(0.25) = ?

Find the inverse Laplace transform f(t) of F(s) = 12/(s + 2) + 10/s with a table of transform pairs, and evaluate f(0.25).

F(s) = 12/(s + 2) + 10/st = 0.25 s
f(0.25) =
2.Easy
t (s)f(t)f(0⁺) = ?

A signal f(t) has the Laplace transform F(s) = (5s² + 10s + 41)/(s(s + 4)(s + 8)). Use the initial value theorem to find f(0⁺), the value of f(t) just after t = 0.

F(s) = (5s² + 10s + 41)/(s(s + 4)(s + 8))
f(0⁺) =
3.Medium
ReIm−1−2−3−4−5−6−7−8−91

F(s) = (5s + 19)/(s² + 9s + 8). Its inverse transform has the form f(t) = A·e−t + B·e−8t. Find the coefficient A by partial fractions.

F(s) = (5s + 19)/(s² + 9s + 8)
A =
4.Medium
ReIm−5−10−15−20−25−30

F(s) = (s + 30)/(s² + 11s + 30). Its inverse transform has the form f(t) = A·e−5t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (s + 30)/(s² + 11s + 30)
A =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 9 and y′(0) = −15 using the Laplace transform, and find y(0.2).

y″ + 5y′ + 6y = 0y(0) = 9y′(0) = −15t = 0.2 s
y(0.2) =
6.Hard
ReIm−0.5−1−1.5−2−2.5−30.51

A system is described by the differential equation y″ + 6y′ + 10y = 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 = 0.2 s. Find g(0.2).

y″ + 6y′ + 10y = 9ut = 0.2 s
g(0.2) =

Answer key

  1. 1. f(0.25) = 17.28
  2. 2. f(0⁺) = 5
  3. 3. A = 2
  4. 4. A = 25
  5. 5. y(0.2) = 6.397
  6. 6. g(0.2) = 0.9813

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