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

Explanation

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.