Q80 / 170 · Manipal (MET) 2015 · Mathematics

Solution of the differential equation $x=1+x y \frac{d y}{d x}+\frac{(x y)^2}{2!}\left(\frac{d y}{d x}\right)^2+\frac{(x y)^3}{3!}\left(\frac{d y}{d x}\right)^3$ $+\ldots$ is

  • A
    $y=\log _\theta(x)+C$
  • B
    $y=\left(\log _e x\right)^2+C$
  • Correct answer
    C
    $y= \pm \sqrt{\left(\log _e x\right)^2+2 C}$
  • D
    $x y=x^y+C$

Solution

We have, $x=e^{x y \frac{d y}{d x}}$ $\begin{array}{ll}\Rightarrow & \log x=x y \frac{d y}{d x} \\ \Rightarrow & y d y=\frac{\log x}{x} d x \\ \Rightarrow & y d y=\log x d(\log x)\end{array}$ On integrating both sides, we get $\frac{y^2}{2}=\frac{\left(\log _\theta x\right)^2}{2}+C$ $\begin{array}{ll}\Rightarrow & y^2=\left(\log _\theta x\right)^2+2 C \\ \Rightarrow & y= \pm \sqrt{\left(\log _\theta x\right)^2+2 C}\end{array}$

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

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