The order of a differential equation is the huge derivative there in the differential equation. Under the differential equations as listed above in equation (3) is a first order differential equation (4), (5), (6), (8) and (9) are second order differential equations, and equation (10) is a third order differential equation and equation (7) is a fourth order differential equation.
Remember that the order doesn't base on whether or not you've obtained ordinary or partial derivatives in the differential equation.
We will be seems almost exclusively at first and second order differential equations in these notes. When you will notice most of the solution techniques for second order differential equations can be simply and naturally extended to higher order differential equations and we'll discuss that notion later on.