Find the particular solutions of the following differential equation :
(1) ` y(1+ log x) dx/dy - x log x = 0 , when , x =e , y= e^(2)`

Options

A. y=ex log x
B. ey=x log x
C. xy=e log x
D. y logx=ex

Answer

A

Explanation

Given differential equation is `y(1+logx)(dx)/(dy)-xlog x=0`
`Rightarrow ((1+logx)dx)/(xlog x)=(dy)/(y)`
`Rightarrow ((1)/(x log x)+(1)/(x))dx=(1)/(y)dy`
On integration both sides, we get
`int ((1)/(x log x)+(1)/(x))dx=int (1)/(y)dy`
Put log x=t
`Rightarrow (1)/(x)dx=dt`
`therefore int (1)/(t)dt+int (1)/(x)dx=int (1)/(y)dy`
`Rightarrow log t+log x=log y+log c`
`Rightarrow log tx=log yc`
`Rightarrow x log x=yc`
When x=e and `y=e^(2)`
`therefore e log e=e^(2)c`
`Rightarrow c Rightarrow (1)/(e)`
`threfore xlogx=(y)/(e)`
`Rightarrow y=ex log x`