- #1
2h2o
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Homework Statement
Find a general solution.
Homework Equations
[tex] 2x\frac{dy}{dx}+y^{3}e^{-2x}=2xy[/tex]
The Attempt at a Solution
Looks like a Bernoulli equation to me, after some algebra:
[tex]\frac{dy}{dx}+\frac{y^{3}}{2xe^{2x}}=y[/tex]
[tex]\frac{dy}{dx}+\frac{y}{2xe^{2x}}=y^{-1}[/tex]
so with [tex]n=-1[/tex]
[tex]v=y^{2}, y=v^{1/2}, \frac{dy}{dx}=\frac{1}{2}v^{-1/2}\frac{dv}{dx}[/tex]
[tex]\frac{1}{2}v^{-1/2}}\frac{dv}{dx}+\frac{v^{1/2}}{2xe^{2x}}=v^{-1/2}[/tex]
[tex]\frac{1}{2}\frac{dv}{dx}+\frac{v}{2xe^{2x}}=1[/tex]
[tex]\frac{dv}{dx}+\frac{v}{xe^{2x}}=2[/tex]
Now an integrating factor:
[tex]\mu=exp[\int{x^{-1}e^{-2x}dx}][/tex]
And that's where I get stuck. This doesn't look like any elementary integral I've learned how to solve, and wolfram|alpha gives me something called the "exponential integral" which we haven't been taught. So I've done something wrong, but I don't see it.
Thanks for any insights.