Vector Representations of Solutions

Consider the linear system of differential equations

displaymath224

This system may be rewritten using matrix-notation. Indeed, set

displaymath226,

then the above system is equivalent to the matricial equation

displaymath228.

Using the matrix product, we get

displaymath230.

The matrix

displaymath232

is called the coefficient matrix of the system. Note that the coefficients of the matrix A can be constant or not. The vector function

displaymath236

is called the nonhomogeneous term.

Remark: One may think that the equation above is only valid for linear systems of two equations. However, that is not the case. For example, consider the linear system

displaymath238

Then, in matricial notation, the system is equivalent to

displaymath240,

where

displaymath242.

Equilibrium Points of Homogeneous Linear Systems

Consider the homogeneous linear system

displaymath244

The equilibrium points are given by the equations

displaymath246

Clearly, x=0 and y=0 give a trivial solution. Hence, the function tex2html_wrap_inline252 gives a constant solution to the linear system. We call it the trivial solution. In general, the equilibrium points are the intersection between two lines. Since the two lines intersect, they are the same (if parallel) or the intersection is reduced to one point. So, the set of equilibrium points is the entire line ax+by=0, or the trivial point (0,0). This conclusion is related to the determinant of the matrix coefficient. Indeed, if

displaymath258

is not equal to 0 (zero), then we have one equilibrium point (the trivial one).

The Linearity Principle

This is may be the most important property for linear systems. Consider the homogeneous linear system

displaymath260,

then

1.
if Y(t) is a solution and k is a constant, then k Y(t) is also a solution;
2.
if tex2html_wrap_inline268 and tex2html_wrap_inline270 are two solutions, then tex2html_wrap_inline272 is also a solution.

This clearly implies that if tex2html_wrap_inline268 and tex2html_wrap_inline270 are two solutions and tex2html_wrap_inline278 and tex2html_wrap_inline280 are two arbitrary constants, then

displaymath282

is also a solution. This conclusion is also known as the Principle of Superposition.

Clearly, from the Principle of Superposition, we may generate plenty of solutions once two solutions are known. The natural question to ask therefore, is whether we have obtained all the solutions. In order to better appreciate this problem let's consider the following example.

Example: Consider the linear system

displaymath284

Show that any solution Y to this system is given as

displaymath282,

where

displaymath290

,

and tex2html_wrap_inline278 and tex2html_wrap_inline280 are two constants.

Answer: It is easy to check that indeed tex2html_wrap_inline268 and tex2html_wrap_inline270 are solutions to the given system. Let Y be any solution. Set

displaymath302.

By the uniqueness and existence theorem, Y is the only solution to the IVP

displaymath306.

Let us find tex2html_wrap_inline278 and tex2html_wrap_inline280 such that tex2html_wrap_inline312 . If this is the case, we should have tex2html_wrap_inline314, which gives

displaymath316,

which implies

displaymath318

Clearly, this gives

displaymath320.

Consider the function

displaymath322.

The linearity principle implies that tex2html_wrap_inline324 is a solution. And, since

displaymath326,

the uniqueness and existence theorem implies that in fact tex2html_wrap_inline328 gives the desired conclusion.

Remark: When you look at the above example you will notice that what made the conclusion work is that we were able to solve the algebraic system

displaymath316

and this was possible because the two vectors

displaymath332

are linearly independent. In fact, the above conclusion is always valid whenever we have a linear independence around.

Theorem: The General Solution

Suppose tex2html_wrap_inline268 and tex2html_wrap_inline270 are two solutions to the linear system

displaymath260.

Assume that the vectors tex2html_wrap_inline340 and tex2html_wrap_inline342 are linearly independent. Then, the solution to the IVP

displaymath344,

is given by

displaymath282,

for some constants tex2html_wrap_inline278 and tex2html_wrap_inline280 . In this case, the two-parameter family

displaymath352,

where tex2html_wrap_inline278 and tex2html_wrap_inline280 are arbitrary constants, is called the general solution of the system. Then, the two solutions tex2html_wrap_inline268 and tex2html_wrap_inline270 are said to be linearly independent.

Example: Consider the undamped harmonic oscillator

displaymath358.

Show that any solution x is given by

displaymath362.

Answer: Consider the associated linear system

displaymath364

Set tex2html_wrap_inline366 . Note that the second component is just the derivative of the first one. Consider the two vector functions

displaymath368

It is easy to check that these two vector functions are in fact solutions to the given system. Also, you may check that the two vectors

displaymath370

are linearly independent. Therefore, any solution Y of the system is given by

displaymath282,

where tex2html_wrap_inline278 and tex2html_wrap_inline280 are two constants. Using the first component of Y, we see that any solution x(t) of the equation is given by

displaymath362,

where tex2html_wrap_inline278 and tex2html_wrap_inline280 are two arbitrary constants.

Clearly, the main problem now is how to find the two linearly independent solutions. This problem will be discussed in the next section and will use the eigenvalue and eigenvector technique.

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour