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

  • question_answer
    Under which one of the following conditions does the solution of \[\frac{dy}{dx}=\frac{ax+b}{cy+d}\] represent a parabola?

    A) \[a=0,\text{ }c=0\]

    B) \[a=1,\text{ }b=2,\text{ }c\ne 0\]

    C) \[a=0,c\ne 0,b\ne 0\]

    D) \[a=1,c=1\]

    Correct Answer: C

    Solution :

    [c] Given: \[\frac{dy}{dx}=\frac{ax+b}{cy+d}\] or \[(cy+d)dy=(ax+b)dx\] Integrating both the sides. \[c.\int{ydy+d\int{dy=a\int{xdx+b\int{dx+K}}}}\] [K is constant integration] or, \[c.\frac{{{y}^{2}}}{2}+d.y=a\frac{{{x}^{2}}}{2}+b.x+K\] or, \[c{{y}^{2}}+2d.y=a{{x}^{2}}+2b.x+2K\] This equation will represent a parabola when either, the coefficient of \[{{x}^{2}}\] or the coefficient of \[{{y}^{2}}\]is zero, but not both. Thus either c = 0 or a = 0 but not both. From the choice given, a = 0, \[c\ne 0\] and \[b\ne 0\].

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