Differential Equations Question 259
Question: If $ y=y(x) $ and $ \frac{2+\sin x}{1+y}( \frac{dy}{dx} )=-\cos x,y(0)=1, $ then $ y( \frac{\pi }{2} ) $ equals
Options:
A) 1/3
B) 2/3
C) -1/3
D) 1
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \frac{dy}{dx}( \frac{2+\sin x}{1+y} )=-\cos x,y(0)=1 $
$ \Rightarrow \frac{dy}{(1+y)}=\frac{-\cos x}{2+\sin x}dx $ Integrating both sides
$ \Rightarrow ln(1+y)=-ln(2+sinx)+C $ Put x = 0 and y = 1
$ \Rightarrow ln(2)=-ln2+C\Rightarrow C=ln4 $ Put $ x=\frac{\pi }{2} $
$ ln(1+y)=-ln3+ln4=ln\frac{4}{3}\Rightarrow y=\frac{1}{3} $
BETA
