Solving Systems of Linear Differential Equations

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

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.
Calf in a field of 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:

The equations to be solved.

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 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:

Finding the characteristic equation.

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:

Calculation for the first eigenvector.

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

The first eigenvector.

For λ = -2, the eigenvector calculation is:

Calculation for the second eigenvector.

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

The second eigenvector.

We now have a solution! In general, the solution is

The general solution.

where λ1 is -3 and λ2 is -2.

Substituting our eigenvalues and eigenvectors:

The solution for our system.

We can now write the x and y separately:

x and y separately.

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