- #1
jbord39
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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?