Question

Determine the differential equation of parabolas with foci at origin and axes along X-axis.
Hint:$y^2=4a\left(x-a\right)$

• A. $y^3{\left(\frac{dy}{dx}\right)}^2=2xy\frac{dy}{dx}-1$
• B. $y^4{\left(\frac{dy}{dx}\right)}^2=2xy\frac{dy}{dx}-1$
• C. $y^3{\left(\frac{dy}{dx}\right)}^2=2xy\frac{dy}{dx}-2$
• D. $y^4{\left(\frac{dy}{dx}\right)}^3=2xy\frac{dy}{dx}-1$

Answer 1

The correct option is B $y^4{\left(\frac{dy}{dx}\right)}^2=2xy\frac{dy}{dx}-1$

Consider the equitation of parabolas with foci at origin and axes along $x-$axis.

$y^2=4a(x-a)$ …….(1)

Differentiate with respect to x,we get

$2y\frac{dy}{dx}=4a(1-0)$

$a=\frac{1}{2y\frac{dy}{dx}}$

Put this value in equation 1^st, we get

$y^2=4\times \frac{1}{2y\frac{dy}{dx}}(x-\frac{1}{2y\frac{dy}{dx}})$

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

$y^3\frac{dy}{dx}=2\left(\frac{2xy\frac{dy}{dx}-1}{2y\frac{dy}{dx}}\right)$

$y^4{\left(\frac{dy}{dx}\right)}^2=2xy\frac{dy}{dx}-1$

Hence ,this is the answer .