Help on Homogeneous Equation

[Scottl]

Hello,

I have a homogeneous equation that I am trying to solve. The original equation is: y'=2xy/x^2-y^2.
To solve I am using v+xdv/dx and y=vx. Here is what I have so far:
v+xdv/dx=2x(vx)/x^2-(vx)^2--> 2x^2v/x^2-x^2v^2= (x^2/x^2)(2v/1-v^2)=-v-2v/1-v^2.
However when I look at the solution it shows: -v(v^2+1)/v^2-1
I am not sure how they received that result.

Thank you

 

[Deleted member 4993]

Hello,

I have a homogeneous equation that I am trying to solve. The original equation is: y'=2xy/x^2-y^2.
To solve I am using v+xdv/dx and y=vx. Here is what I have so far:
v+xdv/dx=2x(vx)/x^2-(vx)^2--> 2x^2v/x^2-x^2v^2= (x^2/x^2)(2v/1-v^2)=-v-2v/1-v^2.
However when I look at the solution it shows: -v(v^2+1)/v^2-1
I am not sure how they received that result.

Thank you

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

\(\displaystyle v + v'x \ = \ \frac{2v}{1-v^2}\)

\(\displaystyle v'x \ = \ \frac{2v}{1-v^2} \ - \ v\)

\(\displaystyle v'x \ = \ \frac{v + v^3}{1-v^2}\)

Continue now.....

 

[yus310]

you can use partial fractions to expand that and then integrate accordingly