# I have the differential equation x <mrow class="MJX-TeXAtom-ORD"> 2 </mr

Nylah Hendrix 2022-07-14 Answered
I have the differential equation
${x}^{2}{y}^{\prime }\left(x\right)+2xy\left(x\right)={y}^{2}\left(x\right)$
With initial condition y(1)=1 and I want to solve this, by observation, we can see the LHS is ${x}^{2}y\left(x\right)$ but I am unfamiliar with tackling the RHS and was wondering where to go from here.
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## Answers (2)

alomjabpdl0
Answered 2022-07-15 Author has 12 answers
This suggests the change of unknown $z={x}^{2}y$, leading to an equation in separeted variables.
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EnvivyEvoxys6
Answered 2022-07-16 Author has 7 answers
First, write the ode as
${x}^{2}{y}^{\prime }\left(x\right)+2xy\left(x\right)={y}^{2}\left(x\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{y}^{\prime }+2\frac{y}{x}=\frac{{y}^{2}}{{x}^{2}}.$
Now, use the change of variables y=xu in the above ode which yields
$x{u}^{\prime }+3u={u}^{2}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\int \frac{du}{{u}^{2}-3u}=\int \frac{dx}{x}.$
I think you can finish it now.
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