QJEEM15Mac4f0: JEE Main 2015 Mathematics - Class 12 JEEM15Mac4f0 Solution: Let y ( x ) be the solution of the differential equation ( x…

Question

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Let

y (x)

be the solution of the differential equation

(x \log x) \frac{d y}{d x} + y = 2 x \log x, (x \geq 1) .

Then

y (e)

is equal to :

Accepted Answer

Given,

\frac{d y}{d x} + (\frac{1}{x \log x}) y = 2

I . F . = e^{\int \frac{1}{x \log x} d x} = e^{\log (\log x)} = \log x

y . \log x = \int 2 \log x d x + c

y \log x = 2 [x \log x - x] + c

Put

x = 1, y .0 = - 2 + c

\Rightarrow c = 2

Put

x = e

y \log e = 2 e (\log e - 1) + c \Rightarrow y (e) = c = 2

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