Substitution for diff. eq.

Apr 2010
11
0
Hi

Having a bit of trouble with what the substition would be to solve this diff. Any ideas? Thanks.

dy/dx = (xy)/(x^2 - y^2)
 
Jul 2007
894
298
New Orleans
Hi

Having a bit of trouble with what the substition would be to solve this diff. Any ideas? Thanks.

dy/dx = (xy)/(x^2 - y^2)
divide the right hand side by x^2

\(\displaystyle \frac{dy}{dx} = \displaystyle{\frac{\frac{xy}{x^2}}{\frac{x^2}{x^2} - \frac{y^2}{x^2}}}\)

\(\displaystyle \frac{dy}{dx} = \displaystyle{\frac{\frac{y}{x}}{1-\bigg(\frac{y}{x}\bigg)^2}}\)

now let

\(\displaystyle v = \frac{y}{x}\)

and

\(\displaystyle vx =y\)

so

\(\displaystyle v + \bigg(\frac{dv}{dx}\bigg)x = \frac{dy}{dx}\)

now sub everything in and solve.