Solve the following differential equation.

Question

anonymous · Answer

$$x^{2} \frac{ dy }{ dx }=x^{2}-xy+y^{2}$$

anonymous · Answer

I'm kinda stuck. I've only learnt how to solve using integrating factors so far.

anonymous · Answer

It doesn't appear to be separable and I can't seem to get it into the proper form to use integrating factors...

anonymous · Answer

it's been a while.  check paul's online math notes.  he does diffy q's in addition to calc.

anonymous · Answer

it is a homogeneous eq.
$$\frac{ dy }{ dx }=\frac{ x^2-xy+y^2 }{ x^2 }$$
put y=vx
$$\frac{ dy }{ dx }=v+x \frac{ dv }{ dx }$$

anonymous · Answer

is v a constant?

anonymous · Answer

nm

anonymous · Answer

Wait, so if you substitute y= vx it's supposed to become your second equation?

anonymous · Answer

$$v+x \frac{ dv }{ dx }=\frac{ x^2-vx^2+v^2x^2 }{ x^2 }=1-v+v^2$$
$$x \frac{ dv }{ dx }=1-2v+v^2=(v-1)^2$$
now separate the variables and integrate.
finally put v=y/x

anonymous · Answer

ah I see now. Thanks :D

anonymous · Answer

v is not constant ,it is a variable.