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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    A solution of the differential equation \[{{\left( \frac{dy}{dx} \right)}^{2}}-x\frac{dy}{dx}+y=0\] is [IIT 1999; Karnataka CET 2002]

    A) \[y=2\]                                      

    B) \[y=2x\]

    C) \[y=2x-4\]                                

    D) \[y=2{{x}^{2}}-4\]

    Correct Answer: C

    Solution :

    • Given equation can be written as \[y=x\frac{dy}{dx}-{{\left( \frac{dy}{dx} \right)}^{2}}\]       
    • If \[\frac{dy}{dx}=p,\]then \[y=px-{{p}^{2}}\]       
    • Differentiating w.r.t. x, we get       
    • \[p=p+x\frac{dp}{dx}-2p\frac{dp}{dx}\]Þ\[\frac{dp}{dx}(x-2p)=0\]Þ\[\frac{dp}{dx}=0\]       
    • Integrating w.r.t. x, we get \[p=c\]       
    • \[\therefore \frac{dy}{dx}=c\]; \[\therefore y=cx-{{c}^{2}}\]                   
    • If \[c=2\], then \[y=2x-4\].


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