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 ( x ) + 2 x y ( x ) = y 2 ( x )
With initial condition y(1)=1 and I want to solve this, by observation, we can see the LHS is x 2 y ( x ) but I am unfamiliar with tackling the RHS and was wondering where to go from here.
x 2 y ( x ) = y 2 ( x )   d x
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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 ( x ) + 2 x y ( x ) = y 2 ( x ) y + 2 y x = y 2 x 2 .
Now, use the change of variables y=xu in the above ode which yields
x u + 3 u = u 2 d u u 2 3 u = d x x .
I think you can finish it now.
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