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February 7th, 2017, 06:29 PM   #1
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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.
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February 7th, 2017, 08:05 PM   #2
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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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