General Contents
Detailed Contents
Index
Programmed tutorial: Solution of a Homogeneous Linear Differential Equation (Second Order): Example 2
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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.
Try factoring this quadratic.
We get
What are the solutions for m?
m
= -3
What do we get for the
general solution
of the original equation?
A linear combination of
e
-3x
and
xe
-3x
.
Write this solution with coefficients.
y = C
1
e
-3x
+ C
2
xe
-3x
Why are there two constants?
The number of constants is always equal to the degree of the differential equation.
How can we check this result?
We can determine the first and second derivatives, and substitute them in the original equation.
Find the first derivative of our solution.
Find the second derivative of our solution.
Substitute these results in the original equation.
becomes
Do the indicated multiplication.
We get
Collect like terms.
We get
Does it check?
Yes.
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