Question

Find the order and degree of the differential equation $\frac{d^{2}y}{d{x}^2}=\sqrt{y+{\left(\frac{dy}{dx}\right)}^2}$.

Answer 1

We need to find the order and degree of the differential equation $\frac{d^{2}y}{d{x}^2}=\sqrt{y+{\left(\frac{dy}{dx}\right)}^2}$

Consider $\frac{d^{2}y}{d{x}^2}=\sqrt{y+{\left(\frac{dy}{dx}\right)}^2}$

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

$\Rightarrow {\left[\frac{d^{2}y}{d{x}^2}\right]}^2=y+{\left(\frac{dy}{dx}\right)}^2$

We know that, the order of a differential equation is the order of the highest order derivative present in the equation and the degree of a differential equation is the power of the highest order derivative in the equation.

Here the highest order derivative is $2$ and its power is $2$.

Thus order is $2$ and power is $2$.