Show that the differential equation `(xsqrt(x^(2)+y^(2)-y^(2)))dx+xydy=0` is homogenous and solve it.

Show that the differential equation `(xsqrt(x^(2)+y^(2)-y^(2)))dx+xydy

1.	Show that the differential equation `(xsqrt(x^(2)+y^(2)-y^(2)))dx+xydy=0` is homogenous and solve it.
Answer» The given differential equation may be written as `(dy)/(dx)=(y^(2)-xsqrt(x^(2)+y^(2))/(xy))`…………….(i) On dividing the Nr and Dr of RHS of (i) by `x^(2)`, we get `(dy)/(dx)={((y/x)^(2)-sqrt(1-(y/x)^(2)))/((y/x))}=f(y/x)`. So, the given differential equation is homogeneous. Putting, `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `v+x(dv)/(dx)=(v^(2)x^(2)-xsqrt(x^(2)+v^(2)x^(2))/(vx^(2)))` `rArr x(dv)/(dx)=((v^(2)-sqrt(1+v^(2)))/(v)-v)` `rArr x(dv)/(dx)=-sqrt(1+v^(2))/(v)` `rArr intv/sqrt(1+v^(2))dy=-int(dx)/x` `rArr sqrt(1+v^(2))=-log\|x\|+C` `rArr sqrt(x^(2)+y^(2))+xlog\|x\|=Cx`, which is the required solution.