Solve the homogenous equation x2xyy2dxx2dy0 Solution x2xyy2d

Solve the homogenous equation (x^(2)-xy+y^(2))dx+x^(2)dy=0

(x^(2)-xy+y^(2))dx+x^(2)dy=0 y= ux ==> u= y/x dy = udx +x du (x^2 - x ux + (ux)^2 ) dx + x^2 (udx+ xdu) = 0 (1 - u + u^2 ) dx + (udx+ xdu) = 0 dx- u dx +u^2 dx + udx + xdu =0 (1+u^2) dx = -x du dx/ x = - du / (1+u^2) ln(x) = - arctan(u) + C Since y= ux ==> u= y/x ln(x) = arctan(y/x) + C