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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The solution of the differential equation \[y-x\frac{dy}{dx}=a\left( {{y}^{2}}+\frac{dy}{dx} \right)\] is [AISSE 1989, 90]

    A) \[y=c(x+a)(1+ay)\]                

    B) \[y=c(x+a)(1-ay)\]

    C) \[y=c(x-a)(1+ay)\]                  

    D) None of these

    Correct Answer: B

    Solution :

    • \[y-x\frac{dy}{dx}=a\left( {{y}^{2}}+\frac{dy}{dx} \right)\] Þ \[y-a{{y}^{2}}=(x+a)\frac{dy}{dx}\]                   
    • Þ \[\frac{dy}{y(1-ay)}=\frac{dx}{x+a}\]                   
    • On integrating both sides, we get                   
    • Þ \[\log y-\log (1-ay)=\log (x+a)+\log c\]                   
    • Þ \[\frac{y}{(1-ay)}=c(x+a)\]or \[c(x+a)(1-ay)=y\].


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