Differential Equations

Taken from Wikipedia.org.

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Examples

In the first group of examples, let u be an unknown function of x, and c and ω are known constants.

\[ {{du}\over{dx}} = cu + x^2 \]

\[ {{d^2 u}\over{dx^2}} - x {{du}\over{dx}} + u = 0\]

\[ {{d^2 u}\over{dx^2}} + \omega^2 u = 0\]

\[ {{du}\over{dx}} = u^2 + 4 \]

\[ L {{d^2 u}\over{dx^2}} + g \sin u = 0 \]

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

\[ {{\partial u}\over{\partial t}} + t {{\partial u}\over{\partial x}} = 0 \]

Solutions

Solving differential equations is not like solving algebraic equations. Not only are their solutions often times unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, it is easy to tell whether a unique solution exists. Given any point (a, b) in the xy-plane, define some rectangular region Z, such that \( Z = [l,m] \times [n,p] \) and (a,b) is in Z. If we are given a differential equation \( {{dx}\over{dt}} = g(x,t) \) and an initial condition \(x(t_0) = x_0\), then there is a unique solution to this initial value problem if \( g(x,t) \) and \( {{\partial g} \over{\partial x}} \) are both continuous on Z. This unique solution exists on some interval with its center at a.

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

\[ f_n(x) \frac{d^n y}{dx^n} + \cdots + f_1(x) \frac{dy}{dx} + f_0 (x)y = h(x) \]

such that

\[ y(x_0) = y_0, y\prime (x_0) = y_0^{\prime}, y\prime\prime (x_0) = y_0^{\prime\prime}, \dots \]

For any nonzero \( f_{n}(x) \), if \( \{f_{0},f_{1},\cdots\} \) and g are continuous on some interval containing \(x_{0}\), y is unique and exists.