Solve the following differential equation: `(x-y)(dy)/(dx)=x+2y`

Answer

`-1/2 log |y^2+xy+x^2|+ sqrt 3 tan^(-1) ((2y+x)/(sqrt 3x))=c`

Explanation

We are given

`(x-y)(dy)/(dx)=x+2y`


On simplifying,

`(dy)/(dx)=(x+2y)/(x-y)-(1)`


now the equation is homogenous as

if `F(x,y)=(x+2y)/(x-y)` then

`F(Ax,Ay)=(Ax+2Ay)/(Ax-Ay)=A(x+2y)/A(x-y)=(x+2y)/(x-y)=A^0F(x,y)`



Now Let us take `y=vx`


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