Ordinary Differential Equations/Linear Systems

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 For example, consider the following first-order linear system of equations: The system of linear equations can be written in matrix form as: So matrix A, is: . The eigenvalues are the diagonal entries in the matrix, and . Because both eigenvalues are real and different, they are categorized as Real Distinct Eigenvalues. The corresponding eigenvectors are: and So the general solution can be written as: Because both eigenvalues are positive, the origin is an unstable node (source). Solutions move away from the origin as . Note, that as the y-component grows, the dy/dt grows faster than dx/dt because grows faster than . As can be seen in the figure below, most trajectories will become increasingly aligned with the y-axis as they move away from x-axis. border|frame|Normalized isotropic vector field (isotropic direction field) of a first-order differential equation with real distinct eigenvalues.|none Complex Eigenvalues A simple example of a linear system with complex eigenvalues is: The matrix for the coefficients for the linear system is: To find the eigenvalues, solve: Thus, Which gives: , thus . So this is a linear system with complex eigenvalues. The slope field consists of circles centered at the origin, with solutions rotating clockwise, as seen in the figure below. none|frame|Vector field (isotropic direction field) of a first-order differential equation with complex eigenvalues. Real Repeated Eigenvalues