Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I have been given this question:

Find the initial value problem of the homogeneous equation:

[tex](x^2 - y^2) y' = xy \ , \ y(1) = 1[/tex]

Now I know, from my lessons, I have to get it in the form of:

[tex]\frac{dy}{dx} = f(\frac{y}{x})[/tex]

I have managed to get close but nothing is working out.

If I make the substitution x = vy then I can get [tex]y' = \frac{v}{1 - v^2}[/tex] but at this state I have made the substitution twice, not once. I then get to a place where I can seperate the variables:

[tex]\int \frac{1 - v^2}{v^3} dv = \int \frac{1}{x} dx[/tex]

[tex]\int \frac{1}{v^3} dv - \int \frac{1}{v} dv = \int \frac{1}{x} dx[/tex]

[tex] \frac{-1}{2v^2} - ln \vert v \vert = ln \vert x \vert + c[/tex]

Then from here my calculations go all funny. Does anyone know if I am barking up the correct tree?

Cheers,

The Bob (2004 ©)

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# Homogeneous Problem

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