3. Solution structure

The rest of this chapter is devoted to linear ODE systems.

Definition 3.8 (Linear system of ODEs)

A linear ODE system is an equation of the form

(3.6)\[\mathbf{x}' = \mathbf{A}(t)\mathbf{x} + \bff(t),\]

where \(\mathbf{x}\) is an \(n\)-dimensional vector, \(\mathbf{A}\) is an \(n\times n\) coefficient matrix, and \(\bff(t)\) is an \(n\)-dimensional forcing function. If the coefficient matrix does not depend on time, the system is said to be constant-coefficient. If given, an initial condition of the system is a time \(t_0\) and vector \(\mathbf{x}_0\) such that \(\mathbf{x}(t_0)=\mathbf{x}_0\).

Example

Here is a basic model for heating in a house. Let \(b(t)\) be the temperature of the basement, \(m(t)\) be the temperature of the main living area, and \(a(t)\) be the temperature of the attic. Suppose the ground is at a constant 10 degrees C. We use a Newtonian model to describe how the temperature of the basement evolves due to interactions with the earth and the main floor:

\[\frac{db}{dt} = -k_b (b - 10) - k_{mb} (b-m). \]

Similarly, the attic interacts with the air, which we will hold at 2 degrees, and the main floor:

\[\frac{da}{dt} = -k_a (a - 2) - k_{ma} (a-m). \]

Finally, suppose the main area interacts mostly with the other levels and experiences input from a heater:

\[\frac{dm}{dt} = -k_{mb} (m - b) - k_{ma} (m - a) + h(t).\]

Write the entire system in the form \(\mathbf{x}' = \mathbf{A} \mathbf{x} + \bff(t)\). That is, identify what \(\mathbf{A}\) and \(\bff(t)\) must be based on the above equations.

Solution

We define \(x_1=b\), \(x_2=m\), and \(x_3=a\). Then

\[ \frac{d\bfx}{dt} = \threevec{(-k_b - k_{mb}) x_1 +k_{mb} x_2}{ k_{mb}x_1 -(k_{mb}+k_{ma})x_2 + k_{ma}x_3}{k_{ma}x_2 -(k_{ma}+k_a)x_3} + \threevec{10k_b}{h(t)}{2k_a}. \]

Observe that on the right side, the terms depending on \(\bfx\) have been separated from the others. Hence

\[\begin{split} \bfx' = \begin{bmatrix} -k_b-k_{mb} & k_{mb} & 0 \\ k_{mb} & -k_{mb}-k_{ma} & k_{ma} \\ 0 & k_{ma} & -k_{ma}-k_a \end{bmatrix} \bfx + \threevec{10k_b}{h(t)}{2k_a}. \end{split}\]

We conclude that

\[\begin{split} \mathbf{A} = \begin{bmatrix} -k_b-k_{mb} & k_{mb} & 0 \\ k_{mb} & -k_{mb}-k_{ma} & k_{ma} \\ 0 & k_{ma} & -k_{ma}-k_a \end{bmatrix}, \quad \bff = \threevec{10k_b}{h(t)}{2k_a}. \end{split}\]

It’s not a stretch to say that virtually all of the general statements we made about the scalar linear problem \(x'=a(t)x+f(t)\) can be remade with boldface/capital letters for the linear system \(\mathbf{x}'=\mathbf{A}(t)\mathbf{x}+\bff(t)\). Those statements relied mainly on linearity. Thus,

Theorem 3.9 (General solutions of first-order linear system)

Every solution of

\[ \mathbf{x}'=\mathbf{A}(t)\mathbf{x}+\bff(t) \]

can be written in the form

\[ \mathbf{x}=\mathbf{x}_h+\mathbf{x}_p, \]

where \(\mathbf{x}_h\) is the general solution of \(\mathbf{x}'=\mathbf{A}(t)\mathbf{x}\) and \(\mathbf{x}_p\) is any solution of \(\mathbf{x}'=\mathbf{A}(t)\mathbf{x}+\bff(t)\).

Once again, then, we look first at the homogeneous system with no forcing term, and then for particular solutions of the original problem.

2. Conversion of higher-order ODEs 4. Homogeneous solutions