June 26th, 2016, 08:08 AM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  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. 
June 26th, 2016, 02:04 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,975 Thanks: 2225 
$xy = u$ 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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