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June 26th, 2016, 08:08 AM   #1
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differential equation

How to solve the differential equation $$x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18 x$$

I think I could let $u=xy$, but I don't know how to proceed it.

Last edited by skipjack; June 26th, 2016 at 02:01 PM.
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June 26th, 2016, 02:04 PM   #2
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$xy = u$ implies $xy' + y = \dfrac{du}{dx}$ and $x\dfrac{d^2y}{dx^2} + 2\dfrac{dy}{dx} = \dfrac{d^2u}{dx^2}$.

Hence $\dfrac{d^2u}{dx^2} + 6\dfrac{du}{dx} + 9u = 18x$.
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