An equation involving derivatives of the dependent variable with respect to independent variable (variables) is known as a differential equation

Examples are $y\frac {dy}{dx} -x=0$, $x\frac {dy}{dx}=1 + x$

Classification of Differential Equations

1. Ordinary Differential Equations (ODEs)
A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.
2. Partial Differential Equations (PDEs)
A PDE is a differential equation that contains unknown multivariable functions and their partial derivatives.

Order and Degree of a Differential Equation

Order
The order of a differential equation is the highest power of the derivative present in the equation.
\[ \frac{d^3y}{dx^3} + x\frac{d^2y}{dx^2} = 0 \]
This equation is of third order.
$\frac {dy}{dx} =e^x$
This equation is of First order.
Degree
The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial equation in derivatives.
\[ \left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0 \]
This equation is of second degree.
\[ \left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^y= 0 \]
The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined