# Solve dy/dx=16xy^2/8x^2y ,the condition given that y(2)=3 . What would be the solution of the equation is y=

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We have given

`(dy)/(dx)=(16xy^2)/(8x^2y) ,and y(2)=3`

`` We can write given differential equation as

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

,since variables are separable ,therefore

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

On itegration ,we have

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

`log(y)=2log(x)+c`

when x=2 ,then y=3

log(3)=2log(2)+c

log(3)-2log(2)=c

log(3/4)=c

Thus

log(y)=2log(x)+lo(3/4)

log(y)=log(x^2)+log(3/4)

log(y)=log(`(3x^2)/4)`

Thus

`y=(3x^2)/4`