Question

The difference between the degree and the order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ is

• A. $2$
• B. $3$
• C. $1$
• D. None of these

Answer 1

The correct option is A

$2$

Explanation for the correct option :

Step-1 : Differentiating the given equation

The given family of curves, $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ $\left(1\right)$

Now, differentiating $\left(1\right)$ with respect to $x$, we get

$2y\frac{dy}{dx}=a$ $\left(2\right)$

Step-2 : Substituting the value of the constants

Now, substituting the value of $a$ from equation $\left(2\right)$ in the equation $\left(1\right)$ i.e. putting $a=2y\frac{dy}{dx}$ in $\left(1\right)$, we obtain

$$\begin{gathered} y^2=a\left(x+\frac{\sqrt{a}}{2}\right) \\ \Rightarrow y^2=2y\frac{dy}{dx}\left(x+\frac{\sqrt{2y{\displaystyle \frac{dy}{dx}}}}{2}\right) \\ \Rightarrow y=2x\frac{dy}{dx}+\sqrt{2y\frac{dy}{dx}}\frac{dy}{dx} \\ \Rightarrow {\left(y-2x\frac{dy}{dx}\right)}^2={\left(\sqrt{2y\frac{dy}{dx}}\frac{dy}{dx}\right)}^2 \\ \Rightarrow y^2-4xy\frac{dy}{dx}+4x^2{\left(\frac{dy}{dx}\right)}^2=2y{\left(\frac{dy}{dx}\right)}^3 \end{gathered}$$

This is the differential equation of the given family of curves.

Step-3 : Calculating difference between the degree and the order of the obtained differential equation.

From Step-2, we can see that the degree of the differential equation is $3$ and the order is $1$.

So, the difference is $3-1=2$.

Hence, option (A) is the correct answer.