Solve: `(dy)/(dx) = (y ( x + 2y))/(x ( 2x + y))` ,y(1) = 2

Explanation

`frac{d y}{d x}=\frac{y(x+2 y)}{x(2 x+y)} y(1)=2 \frac{d y}{d x} `

`frac{y(x+2 y)}{x(2 x+y)}`

It is a homogeneous equation

Put `y=v x`

And `x \frac{d v}{d x}+v=\frac{d y}{d x}`

So,


`x \frac{d v}{d x}+v=\frac{v x(x+2 v x)}{x(2 x+v x)} `

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