(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the following differential equation:

[itex]x^{2}\frac{dy}{dx}=x^{2}-xy+y^{2}[/itex]

State whether this equation is linear or nonlinear and findallit's solutions

2. Relevant equations

I think that the Bernoulli differential equation is relevant, but I'm not sure:

[itex]y'+P(x)y=Q(x)y^{n}[/itex]

3. The attempt at a solution

Ok, so this is really a past exam question, and I've been struggling to remember the method & can't find it anywhere in my notes.

First, I tried to rearrange into something similar to the Bernoulli equation, so I could solve using the method from the wikipedia article (wikipedia dot org slash)wiki/Bernoulli_differential_equation.

The rearrangement I got is

[itex]\frac{y'}{y^{2}}-\frac{1}{y^{2}}+\frac{1}{xy}=\frac{1}{x^{2}}[/itex]

and then I'd use the substitution w=1/y, w'=(-1/y^2)y' but I'm not sure what to do about that pesky -1/y^{2}in the middle. If someone could point me towards the right method to sovle this, I'd be very greatful.

N.B. The course I'm on doesn't cover nonlinear DE's very much, and the only ones that we do have exact solutions. So this should have an exact solution.

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# Homework Help: Nonlinear Differential Equation solving help please!

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