7.5 Homogeneous, Linear Systems with Constant Coefficients

Example (i)

Consider the system of differential equations:

(OverVector[x])'=(({{2, 0}, {0, -1}})) (OverVector[x]).

This system of ODE’s is equivalent to the two equations (Subscript[x, 1])'=2(Subscript[x, 1]) and (Subscript[x, 2])'=-(Subscript[x, 2]).  Since neither of the derivatives depend on the other variable, this is called an uncoupled system.  We can solve each of these equation using techniques from chapter 2: (Subscript[x, 1])(t)=(Subscript[c, 1])(e^(2t)) and (Subscript[x, 2])(t)=(Subscript[c, 2])(e^(-t)).  If we write the solution in a vector form, we have

(OverVector[x])(t)=(Subscript[c, 1])(e^(2t))(({{1}, {0}}))+(Subscript[c, 2])(e^(-t))(({{0}, {1}})).

Note that the eigenvalues and corresponding eigenvectors for the coefficient matrix are:

Eigensystem[{{2,0},{0,-1}}]

In this example, the fundamental solutions each have a form of (e^(λ t))(OverVector[v]), where (λ,(OverVector[v])) are an eigenpair for the coefficient matrix.

Example (ii)

Consider the system of differential equations:

(OverVector[y])'=(({{2, 2}, {1, 3}})) (OverVector[y]).

This system is equivalent to the equations (Subscript[y, 1])'=2(Subscript[y, 1])+2(Subscript[y, 2]) and (Subscript[y, 2])'=(Subscript[y, 1])+3(Subscript[y, 2]).  In this system, since one or more of the derivatives depends on other dependent variables, we say the system is coupled.  We can not use techniques out of chapter 2 to solve these equations, but we can note that if (OverVector[y])(t)=(e^(λ t))(OverVector[v]) is a solution, then

λ (e^(λ t))(OverVector[v])=(OverVector[y])'=(({{2, 2}, {1, 3}})) (OverVector[y])=(({{2, 2}, {1, 3}})) (e^(λ t)) (OverVector[v]) ⇔ (({{2, 2}, {1, 3}})) (OverVector[v])=λ (OverVector[v]).

This mean that (λ,(OverVector[v])) must be an eigenpair of the coefficient matrix.  In this case the eigenvalues and corresponding eigenvectors of the coefficient matrix are:

Eigensystem[{{2,2},{1,3}}]

So the general solution to this system is

(OverVector[y])(t)=(Subscript[c, 1])(e^(4t))(({{1}, {1}}))+(Subscript[c, 2])(e^t)(({{-2}, {1}})).

Final Thoughts

For distinct, real eigenvalues, the solution to the system is as above.  Sections 7.6-7.8 cover the other cases:  for complex eigenvalues and eigenvectors, we use the theorem from 7.4 and find the real and imaginary parts, for repeated eigenvalues it is a bit more complicated (sections 7.7 and 7.8).  We will end the course with section 7.5: real, distinct eigenvalues.