A function y = f (x) satisfies the differential equation dy dx - … - Vidyadip

MCQ

A function $y = f (x)$ satisfies the differential equation $\frac{{dy}}{{dx}} - y = \cos x - \sin x$ with initial condition that $y$ is bounded when $x \rightarrow \infty .$ The area enclosed by $y = f (x), y = \cos x$ and the $y-$ axis is

✓
$\sqrt 2 \; - \;1$
B
$\sqrt 2$
C
$1$
D
$\frac{1}{{\sqrt 2 }}$

✓

Answer

Correct option: A.

$\sqrt 2 \; - \;1$

a
$I.F. = e^{-x}$
$ye^{-x} = \int {{e^{ - x}}(\cos x - \sin x)\,dx} $ put $- x = t$
$= - \int {{e^t}(\cos t + \sin t)\,dt} $
$= - e^t sin t + c$
$y e^{-x} = e^{-x} sin x + c$
since $y$ is bounded when $x \rightarrow \infty \Rightarrow c = 0$
$y = \sin x$
Area $= \int\limits_0^{\frac{\pi }{4}} {(\cos x - \sin x)\,dx} = \sqrt 2 \; - \;1$

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