Here is a system of n differential equations in n unknowns:
This is a constant coefficient linear homogeneous system. Thus, the coefficients are constant, and you can see that the equations are linear in the variables , ..., and their derivatives. The reason for the term "homogeneous" will be clear when I've written the system in matrix form.
The primes on , ..., denote differentiation with respect to an independent variable t. The problem is to solve for , ..., in terms of t.
Write the system in matrix form as
Equivalently,
(A nonhomogeneous system would look like .)
It's possible to solve such a system if you know the eigenvalues (and possibly the eigenvectors) for the coefficient matrix
First, I'll do an example which shows that you can solve small linear systems by brute force.
Example. Consider the system of differential equations
The idea is to solve for and in terms of t.
One approach is to use brute force. Solve the first equation for , then differentiate to find :
Plug these into second equation:
This is a constant coefficient linear homogeneous equation in . The characteristic equation is . The roots are and . Therefore,
Plug back in to find :
The procedure works, but it's clear that the computations would be pretty horrible for larger systems.
To describe a better approach, look at the coefficient matrix:
Find the eigenvalues:
This is the same polynomial that appeared in the example. Since , the eigenvalues are and .
Thus, you don't need to go through the process of eliminating and isolating . You know that
once you know the eigenvalues of the coefficient matrix. You can now finish the problem as above by plugging back in to solve for .
This is better than brute force, but it's still cumbersome if the system has more than two variables.
I can improve things further by making use of eigenvectors as well as eigenvalues. Consider the system
Suppose is an eigenvalue of A with eigenvector v. This means that
I claim that is a solution to the equation, where c is a constant. To see this, plug it in:
To obtain the general solution to , you should have "one arbitrary constant for each differentiation". In this case, you'd expect n arbitrary constants. This discussion should make the following result plausible.
Example. Solve
The matrix form is
The matrix
has eigenvalues and . I need to find the eigenvectors.
Consider :
The last matrix says , or . Therefore,
Take . The eigenvector is .
Now consider :
The last matrix says , or . Therefore,
Take . The eigenvector is .
You can check that the vectors , , are independent. Hence, the solution is
Example. Find the general solution to the linear system
Let
The eigenvalues are and .
For , I have
If is an eigenvector, then
So
is an eigenvector.
For , I have
If is an eigenvector, then
So
is an eigenvector.
The solution is
Example. ( Complex roots) Solve
The characteristic polynomial is
The eigenvalues are . You can check that the eigenvectors are:
Observe that the eigenvectors are conjugates of one another. This is always true when you have a complex eigenvalue.
The eigenvector method gives the following complex solution:
Note that the constants occur in the combinations and . Something like this will always happen in the complex case. Set and . The solution is
In fact, if you're given initial conditions for and , the new constants and will turn out to be real numbers.
You can get a picture of the solution curves for a system even if you can't solve it by sketching the direction field. Suppose you have a two-variable linear system
This is equivalent to the equations
Then
That is, the expression on the right gives the slope of the solution curve at the point .
To sketch the direction field, pick a set of sample points --- for example, the points on a grid. At each point , draw the vector starting at the point . The collection of vectors is the direction field. You can approximate the solution curves by sketching in curves which are tangent to the direction field.
Example. Sketch the direction field for
I've computed the vectors for 9 points:
Thus, from the second line of the table, I'd draw the vector starting at the point .
Here's a sketch of the vectors:
While it's possible to plot fields this way, it's very tedious. You can use software to plot fields quickly. Here is the same field as plotted by Mathematica:
The first picture shows the field as it would be if you plotted it by hand. As you can see, the vectors overlap each other, making the picture a bit ugly. The second picture is the way Mathematica draws the field by default: The vectors' lengths are scaled so that the vectors don't overlap. In subsequent examples, I'll adopt the second alternative when I display a direction field picture.
The arrows in the pictures show the direction of increasing t on the solution curves. You can see from these pictures that the solution curves for this system appear to spiral out from the origin.
Example. ( A compartment model) Two tanks hold 50 gallons of liquid each. The first tank starts with 25 pounds of dissolved salt, while the second starts with pure water. Pure water flows into the first tank at 3 gallons per minute; the well-stirred micture flows into tank 2 at 4 gallons per minute. The mixture in tank 2 is pumped back into tank 1 at 1 gallon per minute, and also drains out at 3 gallons per minute. Find the amount of salt in each tank after t minutes.
Let x be the number of pounds of salt dissolved in the first tank at time t and let y be the number of pounds of salt dissolved in the second tank at time t. The rate equations are
Simplify:
Next, find the characteristic polynomial:
The eigenvalues are , .
Consider :
This says , so . Therefore,
Set . The eigenvector is .
Now consider :
This says , so . Therefore,
Set . The eigenvector is .
The solution is
When , and . Plug in:
Solving for the constants, I obtain , . Thus,
The direction field for the system is shown in the first picture. In the second picture, I've sketched in some solution curves.
The solution curve picture is referred to as the phase portrait.
The eigenvectors and have slopes 2 and -2, respectively. These appear as the two lines (linear solutions).
Consider the linear system
Suppose it has has conjugate complex eigenvalues , with eigenvectors , , respectively. This yields solutions
If is a complex number,
I'll apply this to , using the fact that
Then
The point is that since the terms on the right are independent solutions, so are the terms on the left. The terms on the left, however, are real solutions. Here is what this means.
Example. Solve the system
Set
The eigenvalues are .
Consider :
The last matrix says , so . The eigenvectors are
Take . This yields the eigenvector .
Write down the complex solution
Take the real and imaginary parts:
The general solution is
The eigenvector method produces a solution to a (constant coefficient homogeneous) linear system whenever there are "enough eigenvectors". There might not be "enough eigenvectors" if the characteristic polynomial has repeated roots.
I'll consider the case of repeated roots with multiplicity two or three (i.e. double or triple roots). The general case can be handled by using the exponential of a matrix.
Consider the following linear system:
Suppose is an eigenvalue of A of multiplicity 2, and is an eigenvector for . is one solution; I want to find a second independent solution.
Recall that the constant coefficient equation had independent solutions and .
By analogy, it's reasonable to guess a solution of the form
Here is a constant vector.
Plug the guess into :
Compare terms in and on the left and right:
While it's true that is a solution, it's not a very useful solution. I'll try again, this time using
Then
Note that
Hence,
Equate coefficients in , :
In other words, is an eigenvector, and is a vector which is mapped by to the eigenvector. is called a generalized eigenvector.
Example. Solve
Therefore, is an eigenvalue of multiplicity 2.
Now
The last matrix says , or . Therefore,
Take . The eigenvector is . This gives a solution
Next, I'll try to find a vector such that
Write . The equation becomes
Row reduce:
The last matrix says that , so . In this situation, I may take ; doing so produces .
This work generates the solution
The general solution is
The first picture shows the direction field; the second shows the phase portrait, with some typical solution curves. This kind of phase portrait is called an improper node.
Example. Solve the system
The eigenvalues are and (double).
I'll do first.
The last matrix implies that and , so the eigenvectors are
For ,
The last matrix implies that and , so the eigenvectors are
I'll use .
Next, I find a generalized eigenvector . It must satisfy
That is,
Solving this system yields , . I can take , so , and
The solution is
I'll give a brief description of the situation for an eigenvalue of multiplicity 3. First, if there are three {\it independent} eigenvectors , , , the solution is
Suppose there is one independent eigenvector, say . One solution is
Find a generalized eigenvector by solving
A second solution is
Next, obtain another generalized eigenvector by solving
A third independent solution is
Finally, combine the solutions to obtain the general solution.
The only other possibility is that there are two independent eigenvectors and . These give solutions
Find a generalized eigenvector by solving
The constants a and b are chosen so that the equation is solvable.
yields the solution
The best way of explaining why this works involves something called the Jordan canonical form for matrices. It's also possible to circumvent these technicalities by using the exponential of a matrix.
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Copyright 2015 by Bruce Ikenaga