Show That The Given Differential Equation Is Homogeneous And Solve Each Of Them X2 Y2 Dx 2xy Dy 0 Mathematics Shaalaa Com
Correction (after missing a sign) As kobe pointed out, the original DE is $$ (x^2y^2)y'2xy=0, $$ which as equation for a vector field reads $$ (x^2y^2)\,dy2xy\,dx=0\iff Im(\bar z^2\,dz)=0\text{ with } z=xiy $$ From the complex interpretation it is directly visible that this is not integrable, for that it would have to be an expression $Im(f(z)\,dz)$ But since $\barSo MxNy=4x Now, (MxNy)/N=4x/ (2xy)=2/y, so this is just a function of y v (y) That means that we can let u (y)=e integral of vdy u (y)=e 2ln y =1/y 2 So the equation (x 2 /y 2 1)y'2x/y=0 is exact Now, that means that, letting (x 2 /y 2 1)=fy, 2x/y=fx, then f=x 2 /yyc (x) and f=x 2 /yd (y) These two equations match up, so we
