Let y(x) be the solution of the differential equation `(xlogx)(dy)/(dx)+y=2xlogx, (xge1)`, Then y(e) is equal to

Options

A. 2
B. 2e
C. e
D. 0

Answer

A

Explanation

We have,
`(xlogx)(dy)/(dx)+y=2x logx`
`rArr" "(dy)/(dx)+(1)/(x log x)y=2" …(i)"`
This is a linear differential equation of the form `(dy)/(dx)+Py=Q`,
where `P=(1)/(x log x)and Q=2`.
`"I.F."=e^(intPdx)=e^(int(1)/(xlogx)dx)=e^(log(logx))=logx`
Multiplying both sides of (i) by `"I.F."=logx`, we obtain
`logx(dy)/(dx)+(1)/(x)y=2logx`
Integrating both sides with respect to x, we get
`ylog x=int 2log xdx +C`
`rArr" "ylog x=2x(logx-1)+C" ...(ii)"`
It is given that `x ge1`. Putting x = 1 in (ii), we obtain
`yxx0=2(0-1)+CrArr C=2`
Putting C = 2 in (ii), we get
`ylog x = 2x (log x-1)+2`
Putting x = e, we get y(e) = 2.