In this lesson, we will look at two methods for solving systems of linear differential equations: the eigenvalue method and the Laplace transform method.
Solving Systems of Differential Equations
Imagine a distant part of the country where the life form is a type of cattle we'll call the 'xnay beast' that eats a certain type of grass we'll call 'ystrain grass'. The change in the xnay population depends on the ystrain as well as on the current size of the xnay population. The population of ystrain also depends on xnay and the current amount of ystrain. It's a fascinating mix! The more xnay, the more ystrain gets eaten which reduces the amount of ystrain which can reduce the amount of xnay. Less xnay and the ystrain can thrive. It can be very interesting! Especially for the xnay.
A young calf eating grass.
Having a variable whose rate of change depends on the variable itself, leads to exponential solutions of the differential equation. When we have two or more variables that are also interdependent, we have a system of differential equations and the solution is a mix of exponentials. Population problems are often modeled with systems of differential equations. In this lesson, we will look at two solution methods.
Describing the Equations
There are usually more than two interrelated variables in a population study. Food resources, predators, climate conditions, … will all interact with population size and its rate of change. To keep things simple, we will look at two variables. This will be enough to show the basic ideas of how to solve these systems of equations. For example:
with x(0) = 2 and y(0) = 1.
The little dot over the x and the y on the left-hand side, is the time derivative. The first equation says the time rate of change of x depends on both x and y. The same can be said for the time rate of change of y. It depends on both x and y. Solving these equations tells us how x and y evolve over time. The statement x(0) = 2 means at time t = 0, the population of xnay was 2. The population of ystrain was 1 at time t = 0.
We will use the eigenvalue and the Laplace transform methods to solve this system of equations. You are invited to check out other lessons on linear algebra and Laplace transforms for more details.
Solving Using Eigenvalues
We write the equations in matrix form:
The matrix is called the 'A matrix'. In general, another term may be added to these equations. With no other term, the equations are called homogeneous equations. We will only look at the homogeneous case in this lesson.
The next step is to obtain the characteristic equation by computing the determinant of A - λI = 0. The details:
This tells us λ is -3 and -2. These are the eigenvalues of our system. Sometimes the eigenvalues are repeated and sometimes they are complex conjugate eigenvalues. In our example, we have two distinct and real eigenvalues. We will not cover the other cases in this lesson. Each of these eigenvalues has an eigenvector. For λ = -3, the eigenvector is calculated:
The equation relating a with b is 4a - 3b = 0. We choose a value for one of the letters. For example, letting b = 1 means a = ¾. The eigenvector v1 is
For λ = -2, the eigenvector calculation is:
The resulting equations are 3a - 3b = 0 and 4a - 4b = 0. These equations are true for a = b. Again, we choose a value. If a = 1, then b = 1. The eigenvector v2 is
We now have a solution! In general, the solution is
where λ1 is -3 and λ2 is -2.
Substituting our eigenvalues and eigenvectors:
We can now write the x and y separately:
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We are almost done! The constants c1 and c2 are obtained from the values of x and y at time t = 0. Substituting the t with a 0 gives
Solving for c1 and c2 gives c1 = -4 and c2 = 5. Substituting these values for c1 and c2 gives us
If this example were modeling the xnay, they would become extinct because as t gets larger, x goes to zero. As a check, let's look at another method for solving the same systems of equations. And maybe the xnay beasts will adapt to some additional types of grass food.
Solving Using the Laplace Transform
This second method uses the Laplace transform. Again, if this is unfamiliar material, you are invited to view other lessons for more details.
We start by taking the Laplace transform of both equations:
Substituting for x(0) and y(0):
Solving for X(S) in the first equation, substituting this result for X(S) in the second equation and solving for Y(S):
We use partial fraction expansion to write
Taking the inverse Laplace transform gives us
This agrees with our earlier result. Now to find x(t).
We started with two equations. If we take the second equation and solve by x we get
Substituting for y gives us:
These results agree with the eigenvalue method. It's good to have more than one source for getting an answer. And fortunately, there are other sources of food for the xnay population.
Two or more equations involving rates of change and interrelated variables is a system of differential equations. These systems can be solved using the eigenvalue method and the Laplace transform method.
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