For the given differential equation , find the particular solution satisfying the given condition: `2x y+y^2-2x^2(dy)/(dx)=0; y=2`when x = 1

Explanation

`2xy+y^(2)-2x^(2)(dy)/(dx)=0`
`implies (dy)/(dx)=(2xy+y^(2))/(2x^(2))`……….`(1)`
It is a homogenous differential equation.
Let, `y=vx`
`implies (dy)/(dx)=v+x(dv)/(dx)`
From equation `(1)` ,
`v+x(dv)/(dx)=(2x^(2)v+v^(2)x^(2))/(2x^(2))=v+(1)/(2)v^(2)`
`implies x(dv)/(dx)=(1)/(2)v^(2)`
`implies (2dv)/(v^(2))=(dx)/(x)`
`implies 2int(1)/(v^(2))dv=int(dx)/(x)`
`implies -(2)/(v)+c=logx`
`implies -(2x)/(y)+c=logx`
`implies logx+(2x)/(y)=c` .........`(2)`
Given, `y=2` at `x=1`
`:. log1+(2xx1)/(2)=c`
`implies c=1`
Therefore, the particular solution of the given equation is
`logx+(2x)/(y)=1`