question_answer

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 \[x\to \infty \]. The area enclosed by \[y=f(x),y=cos\,\,x\] and the y-axis is

A) \[\sqrt{2}-1\]

B) \[\sqrt{2}\]

C) 1

D) \[\frac{1}{\sqrt{2}}\]

Correct Answer: A

Solution :

[a] IF \[={{e}^{-x}}\] \[\therefore y{{e}^{-x}}=\int{{{e}^{-x}}(cosx-sinx)dx}\] Put \[-x=t\] \[=-\int{{{e}^{t}}(cost+sint)dt=-{{e}^{t}}\sin t+c}\] \[y{{e}^{-x}}={{e}^{-x}}\sin x+c\] Since, y is bounded when \[x\to \infty \Rightarrow c=0\] \[\therefore y=\sin x\] Area \[=\int_{0}^{\pi /4}{(cosx-sinx)dx=\sqrt{2}}-1\]