Question

Solve the following differential equation.

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

is v a constant?

Answer 7

nm

Answer 8

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

Answer 9

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

Answer 10

ah I see now. Thanks :D

Answer 11

v is not constant ,it is a variable.