Question

The differential equations satisfied by the system of parabolas $y^2=4a\left(x+a\right)$ is:

• A. $y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$
• B. $y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
• C. $y\left(\frac{dy}{dx}\right)-2x\left(\frac{dy}{dx}\right)-y=0$
• D. $y\left(\frac{dy}{dx}\right)-2x\left(\frac{dy}{dx}\right)+y=0$

Answer 1

The correct option is B

$y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$

Explanation for correct option

Finding the differential equation

Given equation, $y^2=4a(x+a).......\left(1\right)$
On differentiating $w.r.t.x$:
$$\begin{gathered} 2y\frac{dy}{dx}=4a \\ \Rightarrow a=\frac{y}{2}\left(\frac{dy}{dx}\right) \end{gathered}$$
On substituting $a$ in equation $\left(1\right)$:
$$\begin{gathered} y^2=4\left(\frac{y}{z}\left(\frac{dy}{dx}\right)\right)(x+a) \\ \Rightarrow y=\frac{2dy}{dx}(x+a) \\ \Rightarrow y=\frac{dy}{dx}\left(x+\frac{y}{2}\frac{dy}{dx}\right) \\ \Rightarrow y=2\left(x\frac{dy}{dx}+\frac{y}{2}{\left(\frac{dy}{dx}\right)}^2\right) \\ \therefore y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0 \end{gathered}$$
Hence, the correct answer is Option (B).