General Contents
Detailed Contents
Index
Programmed tutorial: Solution of a Homogeneous Linear Differential Equation (Second Order): Example 3
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Solve the differential equation
.
General Contents
Detailed Contents
Index
What type of differential equation is this?
It is a linear homogeneous differential equation with constant coefficients.
What is the expected form of the solution?
y =
e
mx
Substitute it in the differential equation.
Do the differentiation.
We get
Factor this result.
We get
Can
e
mx
ever be zero?
No.
What do we conclude from this?
What is the name of this equation?
It is called the
characteristic equation
of the differential equation.
Set up the solution of this equation with the quadratic formula.
We get
What are the solutions for m?
What do we get for the
general solution
of the original equation?
A linear combination of
e3x
cos
(4x)
and
e
3x
sin
(4x)
.
Write this solution with coefficients.
y = C
1
24y = C
1
e
3x
cos
(4x)
+ C
2
e
3x
sin
(4x)
Why are there two constants?
The number of constants is always equal to the degree of the differential equation.
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