Ordinary Differential Equations/Linear Systems

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A system of differential equations is a collection of two or more differential equations, which each ODE may depend upon the other unknown function.

For example consider the equations:

In this case the equation for differential equation for depends on both and . In principle we could also allow to depend on both and , but it is not necessary.

Notice in some cases we find a solution for a system of ODE's. For example in the case above, because doesn't depend on we can solve the second equation (by separating variables or using an integrating factor) to get that . Since there will be a second constant when we solve the first ODE, we choose to call the constant here . Now we can plug this into the first equation to get that: . We can solve this equation by using an integrating factor to get that:

In other cases a clever change of variables allows one to separate the two ODE's. Consider the system


If we let and . Then we find that

and each of these are easy to solve: and . And so we find and . It turns out to be helpful with systems to work with vectors and matrices so if we introduce Then the above system can be re-written as:

And we have solutions and

Notice that the solutions we found were of the for for some constant vector . Using this as motivation we will investigate the question, when does solve the system:

for some constant matrix .

By substituting into the equation we see that:

Since , the only way for the left hand side to be is if is an eigenvalue and is a corresponding eigenvector.

This is not quite the end of the story. When the matrix is real we shall consider the following cases:

Real Distinct Eigenvalues[edit]

Complex Eigenvalues[edit]

Real Repeated Eigenvalues[edit]