Show that the differential equation is `x(dy)/(dx)-y=sqrt(x^(2)+y^(2))`, is homogenous and solve it.

Show that the differential equation is `x(dy)/(dx)-y=sqrt(x^(2)+y^(2))

1.	Show that the differential equation is `x(dy)/(dx)-y=sqrt(x^(2)+y^(2))`, is homogenous and solve it.
Answer» The given differential equation may be written as `(dy)/(dx)=(y+sqrt(x^(2)+y^(2))/(x))`…………………(i) On dividing the Nr and Dr of RHS of (i) by `x`, we get 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)=(vx+sqrt(x^(2)+v^(2)x^(2))/(x))=v+sqrt(1+v^(2))` `rArr x(dv)/(dx)=sqrt(1+v^(2))` `rArr int(dv)/(1+v^(2))=int(dx)/x` `rArr log\|v+sqrt(1+v^(2))\|=log\|C_(1)\|` `rArr (v+sqrt(1+v^(2))/(x))=+-C_(1)=C` (say) `rArr v+sqrt(1+v^(2))=Cx` `rArr y+sqrt(x^(2)+y^(2))=Cx^(2)`, which is the required solution `[therefore v=yx]`.