Differential Equations Question 297
Question: 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=cosx $ and the y-axis is
Options:
A) $ \sqrt{2}-1 $
B) $ \sqrt{2} $
C) 1
D) $ \frac{1}{\sqrt{2}} $
Show Answer
Answer:
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 $
BETA
