Differential Equations

Differential equations is the study of functions are expressed in terms of their own derivatives. While calculus involves computing derivatives and integrals of equations directly, differential equations involves untangling a function from its derivatives in order to find an expression of the function that does not involve its derivatives. For example, the equation $y + 1 = \dfrac{dy}{dx}$ can be solved to be expressed without the derivative: $y = ce^{x} - 1$, where $c$ is the constant of integration.

Many kinds of differential equations have no known solutions or are impossible to solve at all. As a result, most resources on differential equations start by solving highly constrained equations, then slowly removing the constraints to find more general solutions. However, thanks to the magical powers of silicon, computers are able to use numerical methods to compute approximations to many equations, and most resources cover these methods as well.

Differential equations requires a firm knowledge of calculus, attention to detail, and the occasional stroke of genius. However, the subject can be conquered by any mortal.