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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The general solution of \[{{y}^{2}}\,dx+({{x}^{2}}-xy+{{y}^{2}})\,\,dy=0\] is [EAMCET 2003]

    A) \[{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\]

    B) \[2{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log x+c=0\]

    C) \[\log (y+\sqrt{{{x}^{2}}+{{y}^{2}}})+\log y+c=0\]

    D) \[{{\sinh }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\]

    Correct Answer: A

    Solution :

    • \[\frac{dx}{dy}+\frac{{{x}^{2}}-xy+{{y}^{2}}}{{{y}^{2}}}=0\]        
    • \[\frac{dx}{dy}+{{\left( \frac{x}{y} \right)}^{2}}-\left( \frac{x}{y} \right)+1=0\]                   
    • Put \[v=x/y\] Þ \[x=vy\] Þ \[\frac{dx}{dy}=v+y\frac{dv}{dy}\]          
    • \[v+y\frac{dv}{dy}+{{v}^{2}}-v+1=0\] Þ \[\frac{dv}{{{v}^{2}}+1}+\frac{dy}{y}=0\]        
    • Þ  \[\int{\frac{dv}{{{v}^{2}}+1}+\int{\frac{dy}{y}=0}}\] Þ \[{{\tan }^{-1}}(v)+\log y+C=0\]                   
    • Þ  \[{{\tan }^{-1}}(x/y)+\log y+c=0\].


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