A "system" of ODEs (in dimension 2) is an equation of the form
x' = f(x,y,t) y' = g(x,y,t)
(In lecture I used x-dot and y-dot; x and y are functions of t.)
Of special interest is the "linear" case,
x' = a(t)x + b(t)y + r1(t) y' = c(t)x + d(t)y + r2(t)
This admits a matrix formulation: write u = [x;y] (that is, the column vector with entries x, y; in class I write u-arrow), r = [r1;r2], and A is a 2x2 matrix with entries a and b in the top row, and c and d in the bottom row (in Matlab notation, A = [a b; c d] ).
Then our equation is u' = Au + r .
Of special interest as usual is the case in which the "coefficient matrix" A is constant. The equation is "homogeneous" if r = 0.
Model case: the romance of Romeo and Juliet, familiar from Shakespeare in Love. I'll use this to try to convey the dynamics of a linear system. Let x denote the level of R's love for J, and y the level of J's love for R. Suppose the relationship is modeled by x' = 2y, y' = -(1/2)x; or u' = Au, where A = [0 2; -1/2 0]. That is, if J loves R then R's love for J increases, but if R loves J then J's love for R decreases. (Juliet is "fickle.") We can solve this by "elimination": x" = 2y' = -x, so x is a linear combination of cos(t) and sin(t). As is often the case, it's better to use the "polar" parametrization of these solutions: x = A cos(t-phi). Then y = x'/2 = -(A/2)sin(t-phi). The pair (x,y) sweeps out an ellipse, centered at the origin, with major semi-axis A along the x-axis and minor semi-axis A/2 along the y-axis. This "phase portrait" doesn't display the time evolution except that one can denote the arrow of time (clockwise in this case). It's fun to tell the cyclic story of this love affair, quadrant by quadrant. The phase plane is foliated by the "trajectories," which are concentric ellipses (together with 0).
Usually this link between higher order equations and first order systems is traversed in the other direction. For example we had the Airy equation x" + tx = 0. This can be converted into a first order system by introducing an new variable y and declaring that x' = y; then y' = x" = -tx. This is again homogeneous linear, with matrix [0 1; -t 0]. Since Matlab numerical ODE solvers deal only with first order equations, this method is essential in using Matlab.
For another example, suppose x" = 2x' + 3x = 0. Let x' = y, so y' = x" = -3x -2x' = -3x -2y. The matrix is [0 1; -3 -2]. For a third order case, say x''' + ax'' + bx' + cx = 0; x' = y, y' = x'' = z; z' = x''' = - cx - bx' - ax'' = - cx - by - az; now the matrix is [0 1 0; 0 0 1; -c -b -a]. This is the "companion matrix" for the original equation.
Solutions to linear systems are parametrized just as in the case of a single equation. There is exactly one solution with any given initial condition u(a), which however is now a vector. So (in the n = 2 case) there are two independent solutions to a homogeneous linear equation, corresponding for example to the two independent initial conditions u(a) = [1 0] and u(a) = [0 1]. The general solution to u' = Au + r is given by u = up + a1 u1 + ... + an un, where up is any solution and u1,...,un are independent solutions to the homogeneous equation.
In the R&J example (which is homogeneous), we have fundamental solutions
u1 = [cos(t); -(1/2)sin(t)]
u2 = [2 sin(t); cos t] .