Q119 / 160 · AP EAMCET 2022 (08 Jul Shift 2) · Mathematics

Suppose that $f(x, y)$ and $g(x, y)$ are homogeneous functions of same order. If $x=V y$ reduces the equation $\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}$ to the form $\frac{d V}{d y}=\frac{1}{y}(F(V))$, then $F(V)=$

  • A
    $\left(\frac{f(1, V)}{g(1, V)}-V\right)$
  • B
    $\left(\frac{f(V, 1)}{g(V, 1)}-V\right)$
  • C
    $\left(\frac{g(1, V)}{f(1, V)}-V\right)$
  • Correct answer
    D
    $\left(\frac{g(V, 1)}{f(V, 1)}-V\right)$

Solution

$\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}$...(i) $x=V y$ $\begin{array}{ll}\Rightarrow & \frac{d x}{d y}=V+y \frac{d v}{d y} \\ \Rightarrow & y \frac{d V}{d y}=\frac{d x}{d y}-V \\ \Rightarrow & \frac{d V}{d y}=\frac{1}{y}\left(\frac{d x}{d y}-V\right)\end{array}$ From Eq. (i), $\frac{d x}{d y}=\frac{g(x, y)}{f(x, y)}=\frac{g(V y, y)}{f(V y, y)}=\frac{g(V, 1)}{f(V, 1)}$ $\begin{aligned} & \therefore \quad \frac{d V}{d y}=\frac{1}{y}\left(\frac{g(V, 1)}{f(V, 1)}-V\right) \\ & \therefore \quad F(V)=\left(\frac{g(V, 1)}{f(V, 1)}-V\right)\end{aligned}$

Hence, the correct answer is option: 'D'.

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