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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The solution of the differential equation \[\sqrt{a+x}\frac{dy}{dx}+xy=0\]is [MP PET 1998]

    A) \[y=A{{e}^{2/3(2a-x)\sqrt{x+a}}}\]                                    

    B) \[y=A{{e}^{-2/3(a-x)\sqrt{x+a}}}\]

    C) \[y=A{{e}^{2/3(2a+x)\sqrt{x+a}}}\]                                  

    D) \[y=A{{e}^{-2/3(2a-x)\sqrt{x+a}}}\] (Where A is an arbitrary constant.)

    Correct Answer: A

    Solution :

    • Given \[\frac{dy}{dx}+\frac{xy}{\sqrt{a+x}}=0\]Þ\[\frac{dy}{y}=\frac{-xdx}{\sqrt{a+x}}\]       
    • Integrating both sides, \[\int{\frac{dy}{y}}=\int{\frac{-x}{\sqrt{x+a}}dx}\]       
    • \[\log y=-\int_{{}}^{{}}{\frac{x+a-a}{\sqrt{x+a}}}dx\]\[=-\int_{{}}^{{}}{\sqrt{x+a}}dx+\int_{{}}^{{}}{\frac{a}{\sqrt{x+a}}}dx\]                   
    • Þ \[\log y=-\frac{2}{3}{{(x+a)}^{3/2}}+2a\sqrt{x+a}+\log A\]                   
    • \[y=A{{e}^{-2/3{{(x+a)}^{3/2}}+2a\sqrt{x+a}}}\]\[=A{{e}^{\left[ (\sqrt{x+a}\left( -\frac{2}{3}(x+a)+2a \right) \right]}}\]                     
    • \[=A{{e}^{\left[ \sqrt{x+a}\left( \frac{-2x-2a+6a}{3} \right) \right]}}\]\[=A{{e}^{[-2/3\sqrt{x+a}(x-2a)]}}\]                   
    • or \[y=A{{e}^{[2/3\sqrt{x+a}(2a-x)]}}\].


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