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

MCQ

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

✓
$y = \frac{1}{2}(\cos x + \sin x) + c{e^{ - x}}$
B
$y = \frac{1}{2}(\cos x - \sin x) + c{e^{ - x}}$
C
$y = \cos x + \sin x + c{e^{ - x}}$
D
None of these

✓

Answer

Correct option: A.

$y = \frac{1}{2}(\cos x + \sin x) + c{e^{ - x}}$

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

So, $I.F.$ $ = {e^{\int_{}^{} {1dx} }} = {e^x}$

Hence solution is $y.{e^x} = \int_{}^{} {\cos x.{e^x}dx + c} $

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

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