If the solution curve of the differential equation ( y - 2 log e x ) d x + ( x log e x 2 ) d y = 0 , x 1 passes through the points ( e , 4 3 ) and ( e 4 , ) , then is equal to ____ . [2023]

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Solution

Q.
If the solution curve of the differential equation $(y - 2 \log_{e} x) d x + (x \log_{e} x^{2}) d y = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^{4}, α),$ then $α$ is equal to ____ . [2023]

Ans.
(3)

$Given differential equation is$ $(y - 2 \log_{e} x) d x + (x \log_{e} x^{2}) d y = 0$

$\Rightarrow \frac{d y}{d x} = \frac{2 \log_{e} x - y}{x \log_{e} x^{2}}$

$\Rightarrow \frac{d y}{d x} + (\frac{1}{x \log_{e} x^{2}}) y = \frac{2 \log_{e} x}{2 x \log_{e} x} = \frac{1}{x}$

This is a linear differential equation of the type $\frac{d y}{d x} + P y = Q$

$Here, P = \frac{1}{x \log_{e} x^{2}}, Q = \frac{1}{x}$

$I.F. = e^{\int P d x} = e^{\int \frac{1}{2 x \log_{e} x} d x} = \sqrt{\log_{e} x}$

Solution of the differential equation is

$y \sqrt{\log_{e} x} = \int \frac{1}{x} \cdot \sqrt{\log_{e} x} d x$

$\Rightarrow y \sqrt{\log_{e} x} = \frac{2}{3} {(\log_{e} x)}^{3 / 2} + c \dots (i)$

$Now, put (e, \frac{4}{3}) in (i), we get$ $\frac{4}{3} = \frac{2}{3} + c \Rightarrow c = \frac{2}{3}$

$Now, put x = e^{4}, y = α, c = \frac{2}{3} in (i), we get$

$α \sqrt{\log_{e} e^{4}} = \frac{2}{3} {(\log_{e} e^{4})}^{3 / 2} + \frac{2}{3}$

$\Rightarrow α \times 2 = \frac{2}{3} \times 8 + \frac{2}{3} \Rightarrow α = \frac{8}{3} + \frac{1}{3} = 3$

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