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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The solution of the differential equation \[x\frac{dy}{dx}=y(\log y-\log x+1)\]is [IIT 1986; AIEEE 2005]

    A) \[y=x{{e}^{cx}}\]                

    B) \[y+x{{e}^{cx}}=0\]

    C) \[y+{{e}^{x}}=0\]              

    D) None of these

    Correct Answer: A

    Solution :

    • Here \[\frac{dy}{dx}=\frac{y}{x}\left( \log \frac{y}{x}+1 \right)\]                      .....(i)        
    • It is homogeneous equation        
    • So now put \[\]and\[\frac{dy}{dx}=v+x\frac{dv}{dx}\], then the equation (i) reduces to\[\frac{dv}{v\log v}=\frac{dx}{x}\]        
    • On integrating, we get \[\log (\log v)=\log x+\log c\]           
    • Þ \[\log \left( \frac{y}{x} \right)=cx\]Þ\[y=x{{e}^{cx}}\].


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