What is the variation of parameter method for solving differential equations?

[jbord39]

Homework Statement

(D^2 + 2D + 1)y = ln(x)/(xe^x)

Homework Equations

D = d/dx

The Attempt at a Solution

First I find the roots of the left side of the equation, -1 of multiplicity 2.
This leads to
y(c) = Ae^(-x) + Bxe^(-x)

Substituting A and B with a' and b' and dividing both sides by e^(-x) I find the two equations:

-a' + (1-x)b' = ln(x)/x
a' + xb' = 0

Which leads to a' = -ln(x) and b' = ln(x)/x

Integrating by parts leads to:

a = x(1-ln(x))
b = (1/2)[ln(x)]^2

Which leads to

y(p) = ae^(-x) + bxe^(-x)
= xe^(-x)[1-ln(x) + (1/2)(ln(x))^2]

So y = xe^(-x)[1 - ln(x) + (1/2)(ln(x))^2 + A/x + B]

Does this seem correct?

 

[Dick]

It matches what I get. You might notice that the '1' doesn't need to be there, you could just merge it into the constant B.

 

[jbord39]

Thanks a bunch for the reply. I am new to these forums and wondering if there is a "Thank" button or anything like I've seen on other similar forums.

 

[Dick]

Thanks a bunch for the reply. I am new to these forums and wondering if there is a "Thank" button or anything like I've seen on other similar forums.

Not that I know of. But thanks for looking for it.