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Let y(x) be the solution of the differen...

Let `y(x)` be the solution of the differential solution `(xlogx)(dy)/(dx)+y=2xlogx,(x ge1)`. Then y(e ) is equal to

A

(a) e

B

(b) 0

C

(c) 2

D

(d) 2e

Text Solution

Verified by Experts

The correct Answer is:
(c)
Given differential equation is
`(x log x)dy/dx+y=2xlogx`
`rArr dy/dx+y/(xlogx)=2`
This is a linear differential equation.
`therefore IF= e^(int1/(x logx)dx)=e^(log(logx))=log x`
Now, the solution of given differential equation is given
by
`y cdot log x = int log x cdot 2dx`
`rArr y cdot log x = 2 int log x dx`
`rArr y cdot log x = 2 [x log x-x]+c`
At `x = 1 rArr c = 2`
`rArr y cdot log x = 2 [x log x-x]+2`
At `x=e, y=2(e-e)+2`
`rArr y = 2`
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