Question

The differential equation of hyperbola whose axes are along both the axes is $y\frac{dy}{dx}=x{\left(\frac{dy}{dx}\right)}^n+xy\frac{d^{2}y}{d{x}^2}$
Here n =___

Answer 1

We know the standard equations of hyperbola whose axes are along the coordinates axes is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. It is Now since this equation has two constants a and b, we will have to differentiate it twice to get its differential equation. It’s intuitive. Because we have to eliminate 2 parameters and we need 2 equations. So differentiating it we get
$\frac{2x}{a^2}-\frac{2y\frac{dy}{dx}}{b^2}=0$ ...(1)
Differentiating again we get
$\frac{2x}{a^2}-\frac{2}{b^2}({\left(\frac{dy}{dx}\right)}^2+(\frac{d^{2}y}{d{x}^2}\times y))=0$ ...(2)
Equating the values of $\frac{b^2}{a^2}$ from (1) and (2) we get
$\frac{y\frac{dy}{dx}}{x}={\left(\frac{dy}{dx}\right)}^2+\frac{yx{d}^{2}y}{d{x}^2}or\frac{ydy}{dx}=x{\left(\frac{dy}{dx}\right)}^2+xy\frac{d^{2}y}{d{x}^2}$
So n = 2.