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JEE Main & Advanced Mathematics Differential Equations Question Bank Variable Separable type differential equations

  • question_answer
    If \[\left( \frac{2+\sin x}{1+y} \right)\frac{dy}{dx}=-\cos x,\ y(0)=1,\] then \[y\text{ }\left( \frac{\pi }{2} \right)\]= [IIT Screening 2004]

    A)                 1             

    B)                 \[\frac{1}{2}\]

    C)                 \[\frac{1}{3}\]       

    D)                 \[\frac{1}{4}\]

    Correct Answer: C

    Solution :

               The given differential equation is \[\frac{\cos x}{2+\sin x}\,dx+\frac{dy}{y+1}=0\]                    Þ \[\log (2+\sin x)+\log (y+1)=\log c\]                    Þ \[(y+1)\,(2+\sin x)=c\] Þ \[2\times 2=c\] Þ \[c=4\]                                 Thus, \[y+1=\frac{4}{2+\sin x}\]Þ\[y=\frac{2-\sin x}{2+\sin x}\] Þ \[y\,\left( \frac{\pi }{2} \right)=\frac{1}{3}\].


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