KVPY Sample Paper KVPY Stream-SX Model Paper-22

  • question_answer
    Solution of the differential equation \[\frac{dy}{dx}\,-\,y\,=\,\cos \,\,x\,-\,\sin \,x\] satisfying the condition that y should be bounded when \[x\,\to \,+\,\infty \]is

    A) \[y\,=\,\sin \,x\]

    B) \[y\,=\,\,\cos \,x\]

    C) \[y\,=\,\,\sin \,x\,+\,\cos \,x\]

    D) none of these

    Correct Answer: A

    Solution :

    \[\therefore \]      \[y{{e}^{-x}}=\int{{{e}^{-x}}(\cos x-\sin x)}\,dx\]
    \[=\int{{{e}^{-x}}((-1)\sin x+\cos x)}\,dx\]\[\Rightarrow \]            \[{{e}^{-x}}\sin x+\operatorname{c}\]
    \[\therefore \]      \[y=\sin x+c\,\,{{e}^{x}}\]
    Now \[\underset{x\,\to \,\infty }{\mathop{\lim }}\,{{e}^{x}}=\infty \]but since y is bounded.
    \[\therefore \]      \[c=0\]
    \[\therefore \]      \[y=\sin \] is the solution.

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