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February 7th, 2017, 06:29 PM  #1 
Member Joined: Feb 2014 Posts: 84 Thanks: 1  Find the general solution of the differential equation
I'm having trouble finding the general solution of question 3. We just covered this and the teacher hasn't really covered everything but I want the headstart. I tried separating this but it doesn't seem to be working. Could anyone guide me? I wanna look like smart when I go back to class. 
February 7th, 2017, 08:05 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,430 Thanks: 1196 
Try typing it in next time ... it's just a single DE. $x \cdot \dfrac{dy}{dx}=y+2\sqrt{xy}$ $\dfrac{dy}{dx}=\dfrac{y}{x} + 2 \sqrt{\dfrac{y}{x}}$ Let $v = \dfrac{y}{x} \implies y=xv \implies \dfrac{dy}{dx}=x \cdot \dfrac{dv}{dx}+v$ $x \cdot \dfrac{dv}{dx} + v = v + 2\sqrt{v}$ $x \cdot \dfrac{dv}{dx} = 2\sqrt{v}$ $\dfrac{dv}{2\sqrt{v}} = \dfrac{dx}{x}$ $\sqrt{v}=\ln{x}+C$ $v= (\ln{x}+C)^2$ $y=x(\ln{x}+C)^2$ 

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