Show that the differential equation `(y^2-x^2)dy=3xy dx` is homogenous and solve it

Explanation

The given differential equation may be written as
`(dy)/(dx)=(3xy)/(y^(2)-x^(2))`…………..(i)
On dividing the Nr and Dr of RHS of (i) by `x^(2)`, we get
`(dy)/(dx)=(3(y/x))/{(y/x)^(2)-1}=f(y/x)`.
Thus, the given differential equation is homogenous.
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get
`v+x(dv)/(dx)=(3v)/(v^(2)-1)`
`rArr x(dv)/(dx){(3v)/(v^(2)-1)-v}=(4v-v^(3))/(v^(2)-1)`
`rArr (v^(2)-1)/(4v-v^(3))dv=1/xdx`
`rArr int (v^(2)-1)/(v(2-v)(2+v))dv=1/xdx`...........(ii)
Let `(v^(2)-1)/(v(2-v)(2+v))=A/v+B/(2-v)+C/(2+v)`.
Then, `(v^(2)-1)-= A(2-v)(2+v)+Bv(2+v)+Cv(2-v)`.............(iii)
Putting, `v=0` on each side of (iii), we get `A=-1/4`.
Putting v=2 on each side of (iii), we get `B=3/8`.
Putting `v=-2` on each side of (iii), we get `C=-3/8`.
`therefore (v^(2)-1)/(v(2-v)(2+v))=-1/(4v)+3/(8(2-v))-3/(8(2+v))`................(iv)
Putting, these values from (iv) in (ii), we get
`-1/4int(dv)/(v)=3/8int(dv)/(2+v)=int1/xdx`
`rArr int1/xdx+1/4int(dv)/v+3/8int(-dv)/(2-v)=int1/xdx`
`rArr int1/xdx+1/4int(dv)/v+3/8int(-dv)/(2-v)+3/8int(dv)/(2+v)=log|C_(1)|`
`rArr 8log|x|+2log|v|+3log|2-v|+3log|2+v|=8log|C_(1)|`
`rArr |x^(8)v^(2)(2-v)^(3)(2+v)^(3)|=C_(1)^(8)=C` (say)
`rArr |x^(8)v^(2)(2-v)^(3)(2+v)^(3)|=C_(1)^(8)=C` (say)
`rArr x^(8)y^(2)/x^(2)(2-y/x)^(3)(2+y/x)^(3)=C`
`rArr y^(2)(2x-y)^(3)(2x+y)^(3)=C`, where C is an arbirary constant.
`rArr y^(2)(4x^(2)-y^(2))^(2)=C`, which is the required solution.