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 = 1.5f(1.5) = ?

Find the inverse Laplace transform f(t) of F(s) = 20/(s + 1)² with a table of transform pairs, and evaluate f(1.5).

F(s) = 20/(s + 1)²t = 1.5 s
f(1.5) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (2s + 24)/(s² + 8s + 6). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (2s + 24)/(s² + 8s + 6)
f′(0⁺) =
3.Medium
ReIm−2−4−6−8−10−12

F(s) = (3s + 35)/(s² + 7s + 12). Its inverse transform has the form f(t) = A·e−3t + B·e−4t. Find the coefficient A by partial fractions.

F(s) = (3s + 35)/(s² + 7s + 12)
A =
4.Medium
t (s)f(t)t = 0.2f(0.2) = ?

Find f(t) for F(s) = 60/(s(s + 2)(s + 5)) by partial fractions, and evaluate f(0.2).

F(s) = 60/(s(s + 2)(s + 5))t = 0.2 s
f(0.2) =
5.Hard
t (s)y(t)t = 0.2y(0.2) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 7 and y′(0) = −1 using the Laplace transform, and find y(0.2).

y″ + 5y′ + 4y = 0y(0) = 7y′(0) = −1t = 0.2 s
y(0.2) =
6.Hard
ReIm−0.5−1−1.5−20.51

A system is described by the differential equation y″ + 4y′ + 5y = 7u, 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 = 0.6 s. Find g(0.6).

y″ + 4y′ + 5y = 7ut = 0.6 s
g(0.6) =

Answer key

  1. 1. f(1.5) = 6.694
  2. 2. f′(0⁺) = 8
  3. 3. A = 26
  4. 4. f(0.2) = 0.7683
  5. 5. y(0.2) = 6.47
  6. 6. g(0.6) = 1.19

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