Q224 / 270 · NDA 2023 (Phase 1) · Mathematics

Paragraph: Suppose E is the differential equation representing family of curves $y^2=2 c x+2 c \sqrt{c}$ where c is a positive parameter.
Question: What is the order of the differential equation?

  • Correct answer
    A
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $4$

Solution

$y^2=2 c x+2 c \sqrt{c}$ Differentiate wrt $x$, $2 y \frac{d y}{d x}=2 c$ $\Rightarrow \quad c=y \frac{d y}{d x}$ $\Rightarrow \quad y^2=2 x \cdot y \frac{d y}{d x}+2\left(y \frac{d y}{d x}\right)^{3 / 2}$ $\Rightarrow y^2-2 x y \frac{d y}{d x}=2 \cdot\left(y \frac{d y}{d x}\right)^{3 / 2}$ Squaring both sides, $y^4+4 x^2 y^2\left(\frac{d y}{d x}\right)^2-4 x y^3 \frac{d y}{d x}=4 y^3\left(\frac{d y}{d x}\right)^3$ Clearly order $=1$.

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

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