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

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

1.	Show that the differential equation `(dy)/(dx)=(y-x)/(y+x)` is homogenous and solve it.
Answer» Thegiven differential equation is `(dy)/(dx)=(y-x)/(y+x)`. …………….(i) On dividing the Nr and Dr of RHS of (i) by `x`, we get `(dy)/(dx)={(y/x-1)/(y/x+1)}=f(y/x)`. So, the given differential equtions is homogenous. Putting, `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `v+x(dv)/(dx)=(vx-x)/(vx+x)` `rArr v+x(dv)/(dx)=(v-1)/(v+1)` `rArr x(dv)/(dx)=-(1+v^(2))/(1+v)` `rArr x(dv)/(dx)=-1/x dx` `rArr int(1+v)/(1+v^(2))dv=-int(dx)/x`. `rArr int1/(1+v^(2))dv+1/2int(2v)/(1+v^(2))dv=-int(dx)/x`. `rArr tan^(-)v+"log"\|xsqrt(1+v^(2))\|=C` `rArr tan^(-1)v+log\|xsqrt(1+v^(2))\|=C` `rArr tan^(-1)y/x+log\|sqrt(x^(2)+y^(2))\|=C` [Putting `v=y/x`] `rArr tan^(-1)y/x+1/2log(x^(2)+y^(2))=C`, which is the required solution.