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

Explanation

The given differential equation can be expressed as
`(dy)/(dx)=(x+2y)/(x-y)`……………….(i)
On dividing the Nr and Dr of (i) by `x`, we get
`(dy)/(dx)={(1+2.y/x)/(1-y/x)}=f(y/x)`.
So,the given differential equations is homogenous.
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get
`v+x(dv)/(dx) =(1+2v)/(1-v)`
`rArr x(dv)/(dx)={(1+2v)/(1-v)-v}=((1+2v)-v+v^(2))/(1-v)=(v^(2)+v+1)/(1-v)`
`rArr (v-1)/(v^(2)+v+1)dv=-1/xdx rArr int(v-1)/(v^(2)+v+1)dv-int1/xdx`............(ii)
Let `(v-1)=A.d/(dv)(v^(2)+v+1)+B`.
Then, `(v-1)=A(2v+1)+B`.
Comparing coefficients of like powers of v, we get
2A=1 and `A+B=-1` `rArr (A=1/2 "and" B=-3/2`).
Putting `(v-1)=1/2(2v+1)-3/2` in (ii), we get
`int((1/2(2v+1)-3/2)/(v^(2)+v+1))dv-log|x|+C`
`=1/2 int(2v+1)/(v^(2)+v+1)dv-3/2int(dv)/(v^(2)+v+1)=-log|x|+C`
`rArr 1/2log|v^(2)+v+1|-3/2int 1/((v+1/2)^(2)+(sqrt(3)/2)^(2))dv=-log|x|+C`
`rArr 1/2log|v^(2)+v+1|-3/2.2/sqrt(3)tan^(-1)(2v+1)/sqrt(3)-log|x|+C`
`rArr 1/2log{(y^(2)+xy+x^(2))/(x^(2)).x^(2)}=sqrt(3)tan^(-1)(2y+x)/(sqrt(3)x)+C`
`rArr log|x^(2)+xy+y^(2)|=2sqrt(3)tan^(-1)(x+2y)/(sqrt(3)x)+C`, which is the required solution.