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

  • question_answer
    Solution of the differential equation\[x=1+xy\frac{dy}{dx}+\frac{{{x}^{2}}{{y}^{2}}}{2!}{{\left( \frac{dy}{dx} \right)}^{2}}+\]\[\frac{{{x}^{3}}{{y}^{3}}}{3!}{{\left( \frac{dy}{dx} \right)}^{3}}+............\]

    A) \[y=ln\,(x)+c\]

    B) \[y=\,{{(ln\,x)}^{2}}+c\]

    C) \[y=\pm \,ln\,(x)+c\]

    D) \[xy={{x}^{y}}+c\]

    Correct Answer: C

    Solution :

    [c] The given equation is reduced to \[x={{e}^{xy(dy/dx)}}\] \[\Rightarrow \ell nx=xy\frac{dy}{dx}\Rightarrow \int{ydy=\int{\frac{1}{x}\ell nxdx}}\] \[\Rightarrow \frac{{{y}^{2}}}{2}=\frac{{{(\ell nx)}^{2}}}{2}+c\] \[\Rightarrow y=\pm \sqrt{{{(\ell nx)}^{2}}}+c=\pm \ell nx+c\]


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