Question

The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis are respectively

• A. $2,1$
• B. $1,2$
• C. $3,2$
• D. $2,3$

Answer 1

The correct option is B

$1,2$

Explanation for the correct option:

The equation of the parabola whose axis is $X$-axis is $y^2=\pm 4a\left(x-h\right)$

Differentiating the equation with respect to $x$ we have

$$\begin{gathered} \frac{d}{dx}{y}^2=\pm 4a\frac{d}{dx}\left(x-h\right) \\ \Rightarrow 2y\frac{dy}{dx}=\pm 4a\times 1 \\ \Rightarrow y\frac{dy}{dx}=\pm 2a \end{gathered}$$

Again Differentiating with respect to $x$ we have $y^{{\text{'}}^2}+y{y}_2=0$

$$\begin{gathered} \frac{d}{dx}\left(y\frac{dy}{dx}\right)=\frac{d}{dx}\left(\pm 2a\right) \\ \Rightarrow \left(\frac{dy}{dx}\right)\frac{d}{dx}\left(y\right)+y\frac{d}{dx}\left(\frac{dy}{dx}\right)=0\left[\because \frac{d}{dx}\left(uv\right)=v\frac{d}{dx}\left(u\right)+u\frac{d}{dx}\left(v\right)\right] \\ \Rightarrow \left(\frac{dy}{dx}\right)\left(\frac{dy}{dx}\right)+y\frac{d^{2}y}{d{x}^2}=0 \\ \Rightarrow {\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}=0 \end{gathered}$$

Find degree and order:

Order:

The highest order derivative in the given equation is $2$.

Degree:

The power to which the highest order derivative is $1$.

Therefore, the degree of the equation is $1$ and the order is $2$

Hence, option (B) is the correct .