Question

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

Answer 1

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

In this differential equation, the order of the highest order derivative is 2.
Clearly, the R.H.S. of the differential equation cannot be expressed as a polynomial in $\frac{d^{2}y}{d{x}^2}$. So, its degree is not defined.

The order of the differential equation is 2 and its degree is not defined.

It is a non-linear differential equation, as one of its differential co-efficients, that is, $\left(\frac{dy}{dx}\right)$, has exponent 2, which is more than 1.