Differential Equations/Structure of Differential Equations

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Differential equations are all made up of certain components, without which they would not be differential equations. In working with a differential equation, we usually have the objective of solving the differential equation.

Contents

[edit] Differential Equations

The term differential equation was coined by Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y.

However, soon after the first usage of this term, differential equations quickly became understood as any algebraic or transcendental equality which involve differentials or derivatives.

An ordinary differential equation is a relationship between an independent variable, (let us say x), a dependent variable (let us call this y), and the derivatives of the dependent variable y with respect x. Thus, it is any equation of the form

f(x,y,y',y'',y''',...,y(n))

For example:

\frac{dy}{dx}=x \,
\frac{dy}{dx}=2 \sin x^2 \,

are ordinary differential equations. However, they can also involve the derivatives of y in respect to x. For example:

xy\frac{d^2y}{dx^2}+y\frac{dy}{dx}+e^{3x}=0

is also an ordinary differential equation.

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 \,

A partial differential equation is a relationship between two or more dependent variables, one independent variable, and with the derivatives of the dependent variables with respect to the independent variable. Usually they are much more difficult to deal with than ordinary differential equations.

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.

A total differential equation is a relationship between an independent variable, two or more dependent variables, and the derivatives of the dependent variables with respect to the independent variable.

These differential equations can admit solutions where the functions admit a complex argument. These two classes of differential equations 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.

[edit] Fractional Differential Equations

Additionally, fractional differential equations, which may be either ordinary or partial differential equations, also present some peculiarities and for this reason are also studied after a firm grounding in the more usual forms have been mastered.

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 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 generalized functions. Methods for identifying these less well studied functions as solutions to fractional differential equations have yet to be developed systematically.

[edit] Characteristics of Differential Equations

The order of a differential equation is the order of the highest derivative involved in the equation. Thus:

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

is a second-order differential equation, as the highest derivative is the second: d²y/dx².

The degree of a polynomial differential equation is the power to which the highest derivative is raised.

[edit] Solutions of a Differential Equation

A solution of this differential equation is any function y=f(x), which, when substituted into the above equation, satisfies the equation.

An equation of the form

f1(x,y,C1,C2,C3,...,Cn) = 0

with C1,C2,C3,...,Cn as arbitrary constants is called an integral solution of the differential equation if all functions y=f(x) that are solutions to the integral solution when C1,C2,C3,...,Cn are substituted for any values (with the possibility of restrictions) are solutions to the differential equation. Originally, James Bernoulli in 1689 used the term integral and Euler used the term particular integral in 1768. The word solution seems to have first appeared around 1774 by Lagrange, and through Poincaré this term has been established.

A third type of solution is called the parametric solution in the form

x = x(t,C1,C2,C3,...,Cn)

and

y = y(t,C1,C2,C3,...,Cn)

with arbitrary constants C1,C2,C3,...,Cn whenever all functions y=f(x) that make the second equation an identity are also solutions to the differential equation.

People have tried to define general solutions (formerly known as complete integral or complete integral equations due to Euler, these two terms now mean something different) to be integral solutions with arbitrary constants, and singular solutions to be integral solutions which are not contained in the general solution. However, these definitions have turned out to be contradictory, since it may be possible that given one general solution that excludes a singular solution, that another general solution may be found that includes the singular solution. Thus, the idea of singular solutions is contradictory and there is no good way to work with these terms.

Instead, we are going to define general solutions to be an integral solution that includes all solutions of the DE, and a particular solution to be any single solution or integral solution of the DE.

When solving a DE in the crude sense, we aim to find ways to solve equations in particular forms to solutions directly, or to reduce them to a more amenable form. Later, we will aim to solve a DE in a more general sense.

An initial value problem is a differential equation together with the initial conditions that the solution y=f(x) also satisfy the equations

y0 = f(x0)
y1 = f'(x0)
y2 = f''(x0)

...

yn = f(n)(x0)

at a specific x0. If the x0 are different, then it is called a boundary value problem with boundary conditions.

We first consider the simple case of the equation y'=f(x). This is easily solvable with the following theorem that you probably have already proved in Calculus:

Fundamental Theorem of Calculus
Theorem: The solution of the differential equation y'=f(x) (i. e. the antiderivative of the function f(x)) is the definite integral

y=\int_{x_0}^x f(x) dx

Where x0 is an arbitrary constant.

This implies that the anti-derivative takes one arbitrary constant added on to the integral of f(x), and this is usually called C, since changing x0 increases or decreases the function by a constant amount.

Thus, there usually exists an infinity of solutions. 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, there is an infinity of solutions which satisfy the given differential equation y'=f(x). For example:

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

could have

y=x+2 \,

as its answer and still be right. And the general solution to this differential equation of

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

is

y=x+C \,.

In order to get a specific solution, we substitute any value of C.

In order for the differential equation to have only one function as its solution, we usually need initial conditions or boundary conditions.

For example, suppose an conditions of our example above was:

y=1 \mbox{ 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.

[edit] Linear and Non-Linear Differential Equations

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

Linear DEs are the simpler kind. A partial differential equation or an ordinary differential equation that has a degree of 1 and no higher degree is called linear.Thus,

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

is a linear DE.

Non-Linear DEs are much more complex, as they are any DEs that are not linear. For example,

y^2, \,\,\,\sqrt{y}\,\,\,\cos y
\left( \frac{d^2y}{dx^2} \right)^2=-7y
2\frac{d^2y}{dx^2}+y^2=x

are non-linear DEs.

Only a tiny proportion of non-linear DEs are solvable exactly - most have to be approximated.

[edit] 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^2y}{dx^2}-y=0

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

\frac{d^2y}{dx^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^ny}{dx^n}+\ldots+a_1(x)\frac{dy}{dx}+a_0(x)y=0,

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

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

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

[edit] Relationship to other types of equation

The following types of equation are not normally encountered 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.

[edit] 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 will be addressed in the section on the existence and uniqueness theorems that will be proved later.

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 behavior 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.

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