Differential Equations/Structure of Differential Equations

Differential equations are all made up of certain components, without which they would not be differential equations.

Basic Parts of a Differential Equation

The defining characteristic of a DE is the presence of a variable, x, and a dependent variable, i.e. a function of x, usually called y. These two components make up ordinary equations, such as

$y=x$

$y=2\sin x^2$

What sets a differential equation apart is the presence of derivatives of the dependent variable. (They are explained in the Wikipedia derivative article, and also in the Wikibooks text on Calculus). The equations therefore now contain a term, or terms, relating to the rate of change of the dependent variable, y. Please note that there does not have to be a term in y for it to be a DE. A simple DE, therefore, is

$\frac{dy}{dx}=1.$

This DE means that the rate of change of y, i.e. the gradient of the curve $y=f(x)$, is 1. This implies a straight line, $y=x$. Knowing that finding an integral of a function is the reverse of finding the derivative, it is clear that to solve this DE, one must integrate both sides with respect to x. (Note: this may be written I.w.r.t.x. in future).

$\int \frac{dy}{dx}\cdot dx$	$=\int 1\cdot dx$
$y$	$=x$

Boundary Conditions and Concept of Solution

By solving a DE, we aim to find an expression for y in terms of x without recourse to derivatives of x.

As anyone who is familiar with indefinite integration knows, the result above is not the full answer. The graph y=x can be shifted up and down by any amount without changing its gradient, and therefore derivative. In reverse, this means that given a derivative, or function of a derivative, we cannot know exactly which curve it is the derivative of. For example:

$\int \frac{dy}{dx}=\int 1$

could have

$y=x+2$

as its answer and still be right. To overcome this, we add a constant of integration, C to the solution, so

$\int \frac{dy}{dx}=\int 1$

now gives

$y=x+C$

as its solution. This constant can take any value. This is called the general solution of the DE, and is often enough. With the information given, this is as close as we can come to the actual solution. To get closer, we need boundary conditions. Boundary conditions give information about the value of the solution in specified places. For example, suppose the boundary conditions of our example above was:

$y=1\text{ at }x=0$

Having found our general solution, we can now substitute in the boundary conditions to find the constant of integration, C:

$y=x+C$

$1=0+C$

$C=1$

The particular solution is the general solution with the boundary conditions accounted for. Thus,

$y=x+1$

is our particular solution.

To summarize: the general solution is the definition of the family of curves which represent the function that satifies the DE. Particular solutions are the specific solutions to DEs, relating to just one in the family of these functions.

Orders of Differential Equations

The order of a differential equation is the ( order of the ) highest derivative of the dependent variable with a non-zero coefficient. Thus:

$\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}-3y=27x^2$

is a second-order differential equation, as the highest derivative is the second: d²y/dx². Equations of a higher order than the second are generally very difficult to solve.

Linear and Non-Linear Differential Equations

DEs fall into two major types: linear and non-linear.

Linear DEs are the simpler kind. They possess only simple multiples of any derivative of y. Thus,

$\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}-3y=27x^2$

is a linear DE.

Non-Linear DEs are much more complex. They contain complicated functions of y, such as, amongst others,

$y^2,\sqrt{y}\cos y$

Indeed, only a tiny proportion are solvable exactly - most have to be approximated. These contain powers of the derivatives of y. Thus,

${\left(\frac{d^{2}y}{d{x}^2}\right)}^2=-7y$

is a non-linear DE, as is

$2\frac{d^{2}y}{d{x}^2}+y^2=x$

Homogeneous Differential Equations

A homogeneous DE is one in which only the terms involving y ( includes the derivatives of y ) are present in the equation. No terms involving the independent variables must be present in the equation. Therefore:

$\frac{d^{2}y}{d{x}^2}-y=0$

is homogeneous. If something is left over, then the DE is non-homogeneous, like this one:

$\frac{d^{2}y}{d{x}^2}-y=2x$

A constant on the RHS also implies a non-homogeneous DE - after all a constant is still a function.

Generally, if a DE can be written as:

$a_n(x)\frac{d^{n}y}{d{x}^n}+\dots +a_1(x)\frac{dy}{dx}+a_0(x)y=0,$

where a_n(x), etc are functions of x, it is homogeneous. However, if it can only be written as

$a_n(x)\frac{d^{n}y}{d{x}^n}+\dots +a_1(x)\frac{dy}{dx}+a_0(x)y=b(x),$

where b(x) is a function of x, it is non-homogeneous.

Ordinary and Partial Differential Equations

If a DE has only one independent variable, then it is said to be ordinary.

If a DE has more than one independent variable, then it is a partial DE. For example, if the example DEs above included partial derivaties with respect to x or time, t, then they would be considered partial differential equations. These require different methods to solve, and will not be covered until much later in this book.

Additional types of differential equation exist that are rarely included in first courses on differential equations: differential equations where the functions can take complex values and fractional differential equations. These may be either ordinary or partial differential equations. These two classes of differential equation present some peculiarities and for this reason are studied after a firm grounding in the more usual forms have been mastered.

Matrices may also be the subject of differential equations. Because of the non-commutativity of matrix multiplication, care must be taken with the order of factors while solving these equations. Again these are rarely included as part of a first course in differential equations.

Fractional Differential Equations

Fractional differential equations are rarely mentioned in most text books so a brief note is included here. Typical ordinary differential equations involve integer power of derivatives while fractional differential equations involve any power. This class of equation has been studied almost as long as the other types of differential equation but other than the semi derivative equations - those involving powers of +/- 1/2 - methods for solving them in closed form are not known. Many examples of the diffusion equation - a commonly occurring partial differential equation in physics and chemistry - can be reformulated in terms of a semiderivative equation and solved immediately.

One reason for the difficulties encountered with this type of differential equation is because the range of potential solutions is much larger than those encountered elsewhere. Integer valued derivatives require that a function to be differentiable: only functions of this type can be solutions to a typical differential equation. Fractional derivatives may be applied to completely discontinuous functions and some generalised functions. Methods for identifying these less well studied functions as solutions to fractional differential equations have yet to be developed systematically.

Relationship to other types of equation

The following types of equation are not normally encounted in a first course in differential equations but are included here to illustrate the range of problems where differential equations play a role.

It is possible to formulate equations where the function being sought is part of the integrand. Such equations are known as integral equations. It is a theorem in differential equations that states that virtually any differential equation can be reformulated as an integral equation. Integral equations are normally studied after differential equations have been mastered. In practice it is sometimes the case that the corresponding integral equation may be easier to solve than the original differential equation.

It is also possible to encounter equations which include both derivatives and integrals. These equations may or may not be convertible to either purely differential or integral equations.

Another related area is that of difference equations. These equations involve the formation of derivatives where the denominator is not an infinitely small quantity but one of finite size. Their methods of solution parallel those of differential equations. One major difference in their solutions is the role played by the exponential function in differential equations is often replaced by another value which may be complex.

Equations containing both difference and differential terms are not uncommonly encountered in practice. These may be difficult to solve in closed form.

Differential equations may be formulated for matrices as well as for real and complex numbers. Because matrix multiplication is not in general commutative while solving these equations careful attention to the order of the factors must be paid.

Existence and Uniqueness theorems

As well as attempting to solve a new differential equation it is frequently worthwhile determining if a solution to the equation actually exists and if it does whether the solution is unique. The answers to these questions may be quite technical and are not normally part of a first course. All equations in this book can be assumed to have solutions that are unique.

Since most differential equations cannot be solved in closed form, numerical solutions are of great importance. While the existence theorems may seem to be rather esoteric to the beginner they are of considerable importance when attempting a numerical solution: in practice it is very helpful to know that a solution really does exist before trying to compute it.

A related area is the qualitative behaviour of differential equations. Here we try to determine how the equations will behave near points of interest. Again these often require some familiarity with solving differential equations and are normally part of a second course.