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

MCQ

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

A
$y\sin x + \cos 2x = 2c$
B
$2y\sin x + \cos x = c$
C
$y\sin x + \cos x = c$
✓
$2y\sin x + \cos 2x = c$

✓

Answer

Correct option: D.

$2y\sin x + \cos 2x = c$

d
(d) $\frac{{dy}}{{dx}} + y\cot x = 2\cos x$

It is linear equation of the form $\frac{{dy}}{{dx}} + Py = Q$

So, $I.F.$ $ = {e^{\int_{}^{} {Pdx} }} = {e^{\int_{}^{} {\cot xdx} }} = {e^{\log \sin x}} = \sin x$

Hence the solution is $y\sin x = \int_{}^{} {2\sin x\cos xdx + c} $

==> $y\sin x = - \frac{1}{2}\cos 2x + c$ ==> $2y\sin x + \cos 2x = c$.

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