A differential equation is any equation that incorporates a free variable x, a function y(x), and various derivatives of y(x). The order of the equation is the highest derivative that appears. If all the derivatives are raised to the nth power, the equation is said to have degree n. One must determine the function y() to "solve" the equation.
For instance, y′ = xy implies y = E½x2. Plug this expression in for y and verify the result.
Although closed form solutions are always preferable, we often have to settle for an expression involving integrals, e.g. y = f(x) + ∫f(t) (t runs from 0 to x). Still other differential equations can only be approximated numerically on a computer.
To grasp the complexity of this subject, realize that even 0 order equations cannot always be solved. Try x = sin(y)×y. There is no simple formula for y as a function of x.
These are called ordinary differential equations, because there is one function of one variable, usually y(x). Partial differential equations involve several functions, and systems of equations with partial derivatives. I know very little about partial differential equations, and as a result, this is the only paragraph you will find on this subject, except for a simple example.
And now, back to ordinary differential equations.