The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 12
2011
Assumed Knowledge
Finding the roots of quadratic equations.
Euler’s formula
e
iθ
= cos
θ
+
i
sin
θ
.
Objectives
(11a)
To be able to write down the auxiliary (characteristic) equation associated with a
secondorder differential equation with constant coefficients.
(11b
) To be able to construct the solutions to such differential equations in terms of real
exponential and trigonometric functions.
Preparatory Questions
1.
Write down the auxiliary (characteristic) equations for each the following secondorder
linear differential equations with constant coefficients, and find their roots:
(
i
)
d
2
y
dt
2
+ 2
dy
dt

8
y
= 0
(
ii
)
d
2
y
dt
2
+ 2
dy
dt

4
y
= 0
(
iii
)
d
2
y
dt
2

9
y
= 0
(
iv
)
d
2
y
dt
2

2
dy
dt
+ 5
y
= 0
(
v
)
d
2
x
dt
2
+ 2
dx
dt
+
x
= 0
Practice Questions
2.
Find the particular solution of Preparatory Question 1 (
i
) with
y
(0) = 0 and
y
′
(0) = 3.
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 Three '11
 various
 Quadratic equation, Trigraph, Elementary algebra, Preparatory Question

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