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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The differential equation of the family of parabolas with focus at the origin and the x-axis as axis is    [EAMCET 2003]

    A) \[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+4x\frac{dy}{dx}=4y\]       

    B) \[-y\,{{\left( \frac{dy}{dx} \right)}^{2}}=2x\frac{dy}{dx}-y\]

    C) \[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+y=2xy\frac{dy}{dx}\]       

    D) \[y\,{{\left( \frac{dy}{dx} \right)}^{2}}+2xy\frac{dy}{dx}+y=0\]

    Correct Answer: B

    Solution :

    • Equation of family of parabolas with focus at \[(0,\,0)\] and x-axis as axis is \[{{y}^{2}}=4a(x+a)\]  - ..(i)        
    • Differentiating (i) with respect to x,        
    • \[2y{{y}_{1}}=4a;\,\,{{y}^{2}}=2y{{y}_{1}}\left( x+\frac{y{{y}_{1}}}{2} \right)\]                   
    • \[y=2x{{y}_{1}}+yy_{1}^{2}\] Þ \[y{{\left( \frac{dy}{dx} \right)}^{2}}+2x\frac{dy}{dx}=y\].


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