Eliminating the arbitrary constants B and C in the expression $y=\frac{2}{3 \mathrm{C}}(\mathrm{C} x-1)^{3 / 2}+\mathrm{B}$, we get

• A

  $x\left[1+\left(\frac{d y}{d x}\right)^2\right]=\frac{d^2 y}{d x^2}$
• Correct answer
  B

  $2 x\left(\frac{d y}{d x}\right) \frac{d^2 y}{d x^2}=1+\left(\frac{d y}{d x}\right)^2$
• C

  $\left(\frac{d y}{d x}\right) \frac{d^2 y}{d x^2}=1$
• D

  $\left(\frac{d y}{d x}\right)^2+1=\frac{d^2 y}{d x^2}$

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