Differential Equations Question 190
Question: A function $ y=f(x) $ satisfies the condition $ f’(x)\sin x+f(x)\cos x=1 $ being bounded when $ x\to 0 $ . If $ l=\int_0^{\pi /2}{f(x)dx} $ , then
Options:
A) $ \frac{\pi }{2}<l<\frac{{{\pi }^{2}}}{4} $
B) $ \frac{\pi }{4}<l<\frac{{{\pi }^{2}}}{2} $
C) $ 1<l<\frac{\pi }{2} $
D) $ 0<l<1 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \sin x\frac{dy}{dx}+y\cos x=1 $
$ \frac{dy}{dx}+y\cot x=\cos ecx $ IF $ ={e^{\int{\cot xdx}}}={e^{ln(sinx)}}=\sin x $
$ y\sin x=\int{\cos ecx.\sin xdx=x+C} $ IF $ x=0,y $ is finite
$ \therefore C=0 $
$ y=x(cosecx)=\frac{x}{\sin x} $ Now, $ l<\frac{{{\pi }^{2}}}{4}andl>\frac{\pi }{2} $ Hence, $ \frac{\pi }{2}<l<\frac{{{\pi }^{2}}}{4} $
BETA
