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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The solution of the differential equation \[\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \,\left( \frac{y}{x} \right)}{{\varphi }'\,\left( \frac{y}{x} \right)}\] is [DCE 2002]

    A) \[\varphi \,\left( \frac{y}{x} \right)=kx\]                                    

    B) \[x\,\varphi \,\left( \frac{y}{x} \right)=k\]

    C) \[\varphi \,\left( \frac{y}{x} \right)=ky\]                                    

    D) \[y\,\varphi \left( \frac{y}{x} \right)=k\]

    Correct Answer: A

    Solution :

    • \[\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \,\left( \frac{y}{x} \right)}{{\varphi }'\,\left( \frac{y}{x} \right)}\]. Put \[y=vx\]  Þ  \[\frac{dy}{dx}=v+x\frac{dv}{dx}\]        
    • \ The given differential equation becomes        
    • \[v+x\frac{dv}{dx}=v+\frac{\varphi \,(v)}{{\varphi }'\,(v)}\] Þ \[\frac{{\varphi }'(v)}{\varphi (v)}dv=\frac{dx}{x}\]           
    • Þ \[\log \varphi (v)=\log x+\log k\]Þ  \[\varphi (v)=kx\] Þ \[\varphi \,\left( \frac{y}{x} \right)=kx\].


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