PlanetPhysics/Differential Equations

General Remarks.
A differential equation is any equation involving derivatives of a dependent variable with respect to one or more independent variables. Thus

$$ \frac{dy}{dx} = y - x $$

$$ \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 5y = sin x $$

$$ \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 $$

$$ \left( \frac{dy}{dx} + y\right)^2 + x = e^y $$

are classified as differential equations. Equation (3) is called a partial differential equation for obvious reasons. The others are called ordinary differential equations. One may have a system of differential equations involving more than one dependent variable,

$$\frac{d^2x}{dt^2} +\frac{dy}{dy} = t$$ $$ \frac{dx}{dt}-y = 0 $$

In addition to their own mathematical interest differential equations are particularly important since the scientist attempts to describe the behavior of certain aspects of the Universe in terms of differential equations. We list a few of this type.

$$ m\frac{d^2x}{dt^2} + k\frac{dx}{dt} + lx = f(x) $$

$$ L\frac{d^2x}{dt^2} +R\frac{dx}{dt} + \frac{1}{C}x = A \cos(\omega t) $$

$$ \frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2} $$

$$ \frac{d^2 u}{dx^2} +\frac{8 \pi^2 m}{h^2} [E - V(x)]u = 0 $$

$$ R_{ij} = 0 $$

The order of the highest derivative occurring in a differential equation is called the order of the differential equation.

Solution of a Differential Equation. Initial and Boundary Conditions
Let a differential equation be given involving the dependent variable $$\phi$$ and the independent variables $$x$$ and $$y$$. Any function $$\phi(x,y)$$ which satisfies the differential equation is called a solution of the differential equation. For example, it is easy to prove that $$\phi(x,y) = e^x \sin y$$ satisfies

$$\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0.$$

We say that $$e^x \sin y$$ is a solution of $$\nabla^2 \phi = 0$$. It is important to realize that a differential equation has, in general, infinitely many solutions. For example, $$y^{\prime \prime} = 0$$ admits any function $$y = ax +b$$ as a solution, where $$a$$ and $$b$$ are constants which can be chosen arbitrarily. To specify a particular solution, either initial conditions like

$$ y = 2, \,\, y^{\prime} = 1 \,\,\,\,\, when \,\,\, x = 3$$

or boundary conditions like

$$y = 2 \,\,\, when \,\,\, x =3 \,\,\,\,\,\,\, and \,\,\,\,\,\,\, y=4 \,\,\, when \,\,\, x = -1$$

must be given in addition to the differential equation. In the first case $$y = x - 1$$is the solution, and in the second case $$y = -\frac{1}{2}x + \frac{7}{2}$$ is the solution. A full study of a differential equation implies the determination of the most general solution of the equation (involving arbitrary elements which may or may not be constants) and a discussion of how many additional conditions must be imposed in order to fix uniquely the arbitrary elements entering in the general solution. Without attempting an exact statement or proof, at the moment, we state the fact that "in general" the most general solution of an ordinary differential equation of order $$n$$ contains exactly $$n$$ arbitrary constants to be uniquely determined by $$n$$ initial conditions.