JEE Main & Advanced Mathematics Differential Equations Question Bank Self Evaluation Test - Differential Equations

  • question_answer
    What is the solution of the differential equation\[(x+y)(dx-dy)=dx+dy\]?

    A) \[x+y+ln\,\,(x+y)=c\]

    B) \[x-y+ln\,\,(x+y)=c\]

    C) \[y-x+ln\,\,(x+y)=c\]

    D) \[y-x-ln\,\,(x-y)=c\]

    Correct Answer: C

    Solution :

    [c] Differential equation is
    dividing by dx on both the sides
    \[(x+y)\left( 1-\frac{dy}{dx} \right)=1+\frac{dy}{dx}\]
    Putting \[x+y=v\]
    \[1+\frac{dy}{dx}=\frac{dv}{dx}\] and \[\frac{dy}{dx}=\frac{dv}{dx}-1\]
    The equation changes to
    \[v\left\{ 1-\left( \frac{dv}{dx}-1 \right) \right\}=\frac{dv}{dx};\,\,\,v\left( 2-\frac{dv}{dx} \right)=\frac{dv}{dx}\]
    \[\left( \frac{1+v}{v} \right)dv=2dx\] or, \[\left( \frac{1}{v}+1 \right)dv=2dx\]
    Integrating on both the sides.
    \[\log v+v=2x+c\]
    Putting \[v=x+y\]
    \[\log (x+y)+x+y=2x+c\]
    or, \[\log (x+y)+y-x=c\]
    or, \[y-x+\log (x+y)=c\]

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