Question

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

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

Answer 1

Answer: (B)

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

Given $y^2=4a(x+a)$ ....(i)
On differentiating w.r.t $x$ we get
$2y\left(\frac{dy}{dx}\right)=4a$ ....(ii)
Eliminating $a$ Eqs (i) and (ii).
Required differential equation is
$y\{1-{\left(\frac{dy}{dx}\right)}^2\}=2x\left(\frac{dy}{dx}\right)$