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.