Question

The differential equation of the family of curves $y ^ { 2 } = 4 a ( x + a )$ , where a is an arbitrary constant, is

• A. $y \left[ 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right] = 2 x \frac { d y } { d x }$
• B. $y \left[ 1 - \left( \frac { d y } { d x } \right) ^ { 2 } \right] = 2 x \frac { d y } { d x }$
• C. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } = 0$
• D. $\left( \frac { d y } { d x } \right) ^ { 3 } + 3 \frac { d y } { d x } + y = 0$

Answer 1

Answer: (B)

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

Answer 2

Given $y ^ { 2 } = 4 a ( x + a )$ Differentiating, $2 y \left( \frac { d y } { d x } \right) = 4 a$

Eliminating a from (i) and (ii), required equation is

$y \left[ 1 - \left( \frac { d y } { d x } \right) ^ { 2 } \right] = 2 x \frac { d y } { d x }$ .