Question

differential equation of all parabolas whose axis s y - axis.......

• A. y$\frac{dy}{dx}+x\frac{d^{2}y}{d{x}^2}=1$
• B. x$\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=1$
• C. $y^2\frac{dy}{dx}+2x\frac{dy}{dx}=0$
• D. $x\frac{dy}{dx}-x^2\frac{d^{2}y}{d{x}^2}=0$

Answer 1

The correct option is D $x\frac{dy}{dx}-x^2\frac{d^{2}y}{d{x}^2}=0$

Parabola with axis on y- axis is given by,

$x^2=4a(y-c),$ Differentiating, $\{center=(0,c\left)\right\}$

$2x=\frac{4ady}{dx}\Rightarrow a=\frac{x}{2y^{\prime }}$

$\Rightarrow x^2=4\left(\frac{x}{2y^{\prime }}\right)y$ $\Rightarrow x^2{y}^{\prime }=2xy$

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

$\Rightarrow x^2=\frac{4x}{2y^{\prime }}(y-c)$ $\Rightarrow 2y^{\prime }{x}^2=2xy-2cx$

$\Rightarrow y^{\prime \prime }{x}^2+2x{y}^{\prime }=2x{y}^{\prime }+2y-2c$

$\Rightarrow c=\frac{2y-y^{\prime \prime }{x}^2}{2}$

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

$\Rightarrow x^2{y}^{\prime }=x\left(y^{\prime \prime }{x}^2\right)$

$\Rightarrow x\frac{dy}{dx}-x^2\frac{d^{2}y}{d{x}^2}=0$