Question

The differential equation of the family of curve $y^2=4a(x+a)$ is

• A. $y^2=4\frac{dy}{dx}(x+\frac{dy}{dx})$
• B. $y^2{\left(\frac{dy}{dx}\right)}^2+2xy\frac{dy}{dx}-y^2=0$
• C. $y^2\frac{dy}{dx}+4=0$
• D. $2y\frac{dy}{dx}+4a=0$

Answer 1

Answer: (B)

$y^2{\left(\frac{dy}{dx}\right)}^2+2xy\frac{dy}{dx}-y^2=0$

$y^2=4a(x+a)$

Differentiate on both sides

$\Rightarrow 2y{y}^{{}^{\prime }}=4a$

$\Rightarrow a=\frac{y{y}^{{}^{\prime }}}{2}$

Substitute in above parabola equation

$\Rightarrow y^2=2y{y}^{{}^{\prime }}(x+\frac{y{y}^{{}^{\prime }}}{2})$

$\Rightarrow y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$

Multiply y on both sides

$\Rightarrow y^2{\left(\frac{dy}{dx}\right)}^2+2xy\frac{dy}{dx}-y^2=0$