y2 dx + (x2 xy + y2) dy = 0 - Maths -

Question

y 2 dx + (x 2
− xy + y 2) dy = 0

Global Expert · Accepted Answer

We have,y2 dx+x2-xy+y2 dy=0dydx=-y2x2-xy+y2This is a homogeneous differential equation
Putting y=vx and dydx=v+xdvdx, we get v+xdvdx=-v2x2x2-vx2+v2x2⇒v+xdvdx=-v21-v+v2⇒xdvdx=-v21-v+v2-v⇒xdvdx=-v-v31-v+v2⇒1-v+v2v+v3dv=-1xdx⇒1+v2-vv1+v2dv=-1xdxIntegrating both sides, we get ∫1+v2-vv1+v2dv=∫1xdx⇒∫1+v2v1+v2dv-∫vv1+v2dv=-∫1xdx⇒∫1vdv-∫11+v2dv=-∫1xdx⇒log v-tan -1v=-log x+log C⇒log vxC=tan-1v⇒vxC= etan-1vPutting v=yx, we get⇒y=Cetan-1vHence, y=Cetan-1v is the required solution

y 2 dx + (x 2 − xy + y 2) dy = 0

y ² dx + (x ² − xy + y ²) dy = 0