JEE Main & Advanced Mathematics Differential Equations Question Bank Self Evaluation Test - Differential Equations

  • question_answer
    If \[y=y(x)\] and \[\frac{2+\sin x}{1+y}\left( \frac{dy}{dx} \right)=-\cos \,x,y(0)=1,\]then \[y\left( \frac{\pi }{2} \right)\] equals

    A) 1/3

    B) 2/3

    C) -1/3

    D) 1

    Correct Answer: A

    Solution :

    [a] \[\frac{dy}{dx}\left( \frac{2+\sin x}{1+y} \right)=-\cos x,y(0)=1\] \[\Rightarrow \frac{dy}{(1+y)}=\frac{-\cos x}{2+\sin x}dx\] Integrating both sides \[\Rightarrow \,\,\,ln(1+y)=-ln(2+sin\,x)+C\] Put x = 0 and y = 1 \[\Rightarrow \,\,\,ln(2)=-ln2+C\Rightarrow C=\,\,ln\,\,4\] Put \[x=\frac{\pi }{2}\] \[ln\,(1+y)=-ln3+ln4=ln\frac{4}{3}\Rightarrow y=\frac{1}{3}\]

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