The solution of the differential equation x x dy dx + y = 2 x is - Vidyadip

MCQ

The solution of the differential equation $x\log x\frac{{dy}}{{dx}} + y = 2\log x$ is

A
$y = \log x + c$
B
$y = \log {x^2} + c$
✓
$y\log x = {(\log x)^2} + c$
D
$y = x\log x + c$

✓

Answer

Correct option: C.

$y\log x = {(\log x)^2} + c$

c
(c) $x\log x\frac{{dy}}{{dx}} + y = 2\log x$ ==> $\frac{{dy}}{{dx}} + \frac{1}{{x\log x}}y = \frac{2}{x}$

This is linear differential equation in y.

$\therefore $ $I.F.$ $ = {e^{\int_x^{} {\frac{1}{{\log x}}dx} }} = {e^{{{\log }_e}{{\log }_e}x}} = \log x$

==> $y.(I.F.)$$ = \int_{}^{} {Q\;.\;(I.F.)\,dx} $ ==> $y\log x = \int_{}^{} {\frac{2}{x}} .\log xdx$

==> $y\log x = {(\log x)^2} + c$.

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