Question

The differential equation of all parabola's with axis parallel to axis of $y$-axis is:

• A. $y_2=2y_1+x$
• B. $y_3=2y_1$
• C. $y_2^3=y_1$
• D. $y_3=0$

Answer 1

Answer: (D)

$y_3=0$

General equation of parabola parallel to y-axis and having center (h,k) and having distance 'a' from vertex to focus is $(x-h{)}^2=4a(y-k)$

Differentiate the above differential equation once.

$\frac{dy}{dx}(x-h{)}^2=\frac{dy}{dx}(4a(y-k))$

$\Rightarrow 2(x-h)=4a\frac{dy}{dx}$

Differentiating above equation once again.

$\Rightarrow \frac{dy}{dx}\left(2\right(x-h\left)\right)=\frac{dy}{dx}\left(4a\frac{dy}{dx}\right)$

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

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{1}{2a}$

Differentiating the above equation once again.

$\Rightarrow \frac{dy}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)=\frac{dy}{dx}\left(\frac{1}{2a}\right)$

$\Rightarrow \frac{d^{3}y}{d{x}^3}=0$

$\Rightarrow y_3=0$