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

  • question_answer
    If for the differential equation \[y'=\frac{y}{x}+\phi \left( \frac{x}{y} \right),\] the general solution is \[y=\frac{x}{\log \left| Cx \right|},\] then \[\phi (x/y)\] is given by

    A) \[-{{x}^{2}}/{{y}^{2}}\]

    B) \[-{{y}^{2}}/{{x}^{2}}\]

    C) \[{{x}^{2}}/{{y}^{2}}\]

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

    Correct Answer: D

    Solution :

    [d] Putting \[v=y/x\] so that \[x\frac{dv}{dx}+v=\frac{dv}{dx}\] We have \[x\frac{dv}{dx}+v=v+\phi (1/v)\] \[\Rightarrow \frac{dv}{\phi (1/v)}=\frac{dx}{x};\Rightarrow \log \left| Cx \right|=\int{\frac{dv}{^{\phi (1/v)}}}\] But \[y=\frac{x}{\log \left| Cx \right|}\] is the general solution, So \[\frac{x}{y}=\frac{1}{v}=\log \left| Cx \right|=\int{\frac{dv}{\phi (1/v)}}\] \[\Rightarrow \phi (1/v)=-1/{{v}^{2}}\] (Differentiating w.r.t.v both sides) \[\Rightarrow \phi (x/y)=-{{y}^{2}}/{{x}^{2}}\]

You need to login to perform this action.
You will be redirected in 3 sec spinner