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Section 27.3 Concepts

Subsection 27.3.1 Change of variables in a system of first-order, linear differential equations

A system of linear differential equations such as

\begin{equation*} \left\{\begin{array}{rcrcr} \dd{y_1}{t} \amp = \amp a y_1(t) \amp + \amp b y_2(t) \\ \dd{y_2}{t} \amp = \amp c y_1(t) \amp + \amp d y_2(t) \end{array}\right. \end{equation*}

has the same form as a system of linear equations, and we can write it as a single matrix equation as

\begin{equation*} \begin{bmatrix} \dd{y_1}{t} \\ \dd{y_2}{t} \end{bmatrix} = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}\text{.} \end{equation*}

If we treat the differential operator \(\ddt\) as a scalar, we can instead write

\begin{equation*} \ddt \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix} = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}\text{,} \end{equation*}

or even more compactly as

\begin{gather} \dd{\uvec{y}}{t} = A \uvec{y}\text{,}\label{equation-sys-diff-eq-concepts-2x2-diff-matrix-eq}\tag{\(\star\)} \end{gather}

where

\begin{equation*} A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \end{equation*}

is the coefficient matrix, and

\begin{equation*} \uvec{y}(t) = \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix} \end{equation*}

is a vector that collects together the unknown functions \(y_1(t), y_2(t)\text{.}\)

Suppose we use an invertible matrix \(Q\) to change variables by

\begin{equation*} \uvec{w}(t) = Q \uvec{y}(t) \text{.} \end{equation*}

Then the new functions \(w_1(t),w_2(t)\) in the function vector \(\uvec{w}(t)\) will be related to the old functions \(y_1(t),y_2(t)\) via linear formulas, where the coefficients come from the entries of \(Q\text{.}\) And, reversing the change of variables to \(\uvec{y} = \inv{Q} \uvec{w}\text{,}\) we can substitute into our differential matrix equation (\(\star\)) to get a new equation

\begin{gather} \ddt (\inv{Q} \uvec{w}) = A \inv{Q} \uvec{w}\text{.}\label{equation-sys-diff-eq-concepts-2x2-transformed-diff-matrix-eq}\tag{\(\star\star\)} \end{gather}

While we may not have heard it phrased in this language, one of the first things that we learn about the derivative in a first course on differential calculus is that it is a linear operator:

\begin{align*} \ddt \bigl( k y(t) \bigr) \amp = k \dydt \text{,} \amp \ddt \bigl( y_1(t) + y_2(t) \bigr) \amp = \dd{y_1}{t} + \dd{y_2}{t} \end{align*}

(where \(k\) is an arbitrary constant). It follows from these properties that differentiation of the entries in a vector of functions and multiplication by a matrix of constants can be performed in either order:

\begin{equation*} \ddt \bigl( M \uvec{y}(t) \bigr) = M \dd{\uvec{y}}{t} \text{.} \end{equation*}

This lets us rearrange (\(\star\)) as

\begin{equation*} \inv{Q} \dd{\uvec{w}}{t} = A \inv{Q} \uvec{w} \qquad \implies \qquad \dd{\uvec{w}}{t} = Q A \inv{Q} \uvec{w}\text{.} \end{equation*}

We immediately see the similarity relationship between the coefficient matrices of the original differential equation and the transformed one. If we reverse the point of view on the change of variables by setting \(P = \inv{Q}\text{,}\) so that

\begin{equation*} \uvec{y}(t) = P \uvec{w}(t) \text{,} \end{equation*}

then we can write the transformed differential matrix equation as

\begin{equation*} \dd{\uvec{w}}{t} = \inv{P} A P \uvec{w} \text{.} \end{equation*}

Assuming the new transformed system is easier to solve, we can transfer solutions for \(w_1(t),w_2(t)\) back to solutions for \(y_1(t),y_2(t)\) using \(P\text{.}\)

Of course, all of the above can be carried out for larger systems of equations involving more unknown functions as well.

Subsection 27.3.2 Systems with a diagonalizable coefficient matrix

A system of differential equations with a diagonal coefficient matrix is easy to solve. For example, system

\begin{equation*} \begin{bmatrix} \dd{w_1}{t} \\ \dd{w_2}{t} \end{bmatrix} = \begin{bmatrix} \lambda_1 \\ \amp \lambda_2 \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \end{equation*}

is not coupled — it describes two separate differential equations

\begin{align*} \dd{w_1}{t} \amp = \lambda_1 w_1 \text{,} \\ \dd{w_2}{t} \amp = \lambda_2 w_2 \text{,} \end{align*}

where the rate of change of each function is proportional to the value of the function. As we reminded ourselves in Discovery 27.1, such equations have exponential solutions:

\begin{align*} w_1(t) \amp = c_1 e^{\lambda_1 t} \text{,} \\ w_2(t) \amp = c_2 e^{\lambda_2 t} \text{,} \end{align*}

where the \(c_j\) are the initial values of the functions.

If instead we have a system \(\uvec{y}' = A \uvec{y}\) with a diagonalizable coefficient matrix \(A\text{,}\) then we can carry out a change of variables as in Subsection 27.3.1 to obtain transformed system \(\uvec{w}' = \inv{P} A P \uvec{w}\) that has diagonal coefficient matrix \(\inv{P} A P\text{.}\) The solutions to this transformed system are again exponential, but since we know that the diagonal entries in diagonal form \(\inv{P} A P\) are precisely the eigenvalues of \(A\text{,}\) the rate constants in the exponential solutions are the eigenvalues of the coefficient matrix \(A\).

Using the transition matrix \(P\text{,}\) we can transform the solutions for \(w_1(t),w_2(t)\) back to solutions

\begin{align*} y_1(t) \amp = p_{11} c_1 e^{\lambda_1 t} + p_{12} c_2 e^{\lambda_2 t} \text{,} \\ y_2(t) \amp = p_{21} c_1 e^{\lambda_1 t} + p_{22} c_2 e^{\lambda_2 t} \text{.} \end{align*}

The above formulas represent the general solution to the system of equations in arbitrary parameters \(c_1,c_2\text{.}\) To obtain a particular solution, initial values \(y_1(0) = a_1\) and \(y_2(0) = a_2\) might be used. But solving the equations \(y_1(0) = a_1\) and \(y_2(0) = a_2\) for coefficients \(c_1,c_2\) is equivalent to solving

\begin{equation*} P \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \text{,} \end{equation*}

which we can do with \(\inv{P}\text{:}\)

\begin{equation*} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = \inv{P} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \text{.} \end{equation*}

Notice what this is doing: the coefficients \(c_1,c_2\) are the initial conditions of our transformed functions

\begin{align*} w_1(t) \amp = c_1 e^{\lambda_1 t} \text{,} \\ w_2(t) \amp = c_2 e^{\lambda_2 t} \text{,} \end{align*}

and multiplying the vector of initial conditions \(\left[\begin{smallmatrix} a_1 \\ a_2 \end{smallmatrix}\right] \) by the transition matrix \(\inv{P}\) converts these initial conditions for \(y_1(t),y_2(t)\) into initial conditions for \(w_1(t),w_2(t)\text{,}\) which provides an immediate answer for the values of \(c_1\) and \(c_2\text{.}\) See Section 27.4 for an example of solving an initial value problem involving a system of differential equations.

Subsection 27.3.3 Homogeneous, second-order, linear differential equations

Every homogeneous, second-order, linear differential equation

\begin{gather} y'' - \alpha y' - \beta y = 0\label{equation-sys-diff-eq-concepts-2nd-order}\tag{\(\star\star\star\)} \end{gather}

can be recast as a coupled system of differential equations by setting \(x(t) = y'(t)\text{.}\) Then

\begin{equation*} x' = y'' = \alpha y' + \beta y = \alpha x + \beta y \text{,} \end{equation*}

giving us system

\begin{equation*} \left\{\begin{array}{rcrcr} x' \amp = \amp \alpha x \phantom{\text{.}} \amp + \amp \beta y \text{,} \\ y' \amp = \amp x \text{.} \end{array}\right. \end{equation*}

We can attempt to decouple this system by diagonalizing the coefficient matrix, as in Subsection 27.3.2, but if we are able to solve this way we should remember that in the end we are only interested in the solution for \(y(t)\text{,}\) as the solution of \(x(t)\) should just be the derivative of \(y(t)\text{.}\)

Let's take at least the first steps to diagonalizing the coefficient matrix

\begin{equation*} A = \begin{bmatrix} \alpha \amp \beta \\ 1 \amp 0 \end{bmatrix} \end{equation*}

by computing

\begin{equation*} \lambda I - A = \begin{bmatrix} \lambda - \alpha \amp - \beta \\ - 1 \amp \lambda \end{bmatrix} \text{,} \end{equation*}

which leads to characteristic polynomial

\begin{equation*} c_A(\lambda) = \det (\lambda I - A) = \lambda^2 - \alpha \lambda - \beta \text{.} \end{equation*}

If you compare this characteristic polynomial with the original second-order differential equation (\(\star\star\star\)), you should notice a similarity.

In fact, every system of equations can be converted to a homogeneous, linear, second-order differential equation, and the above pattern between characteristic polynomial and original second-order equation will recur. For system

\begin{equation*} \left\{\begin{array}{rcrce} x' \amp = a x + b y \text{,} \\ y' \amp = c x + d y \text{,} \end{array}\right. \end{equation*}

we can compute

\begin{equation*} x'' = a x' + b y' = a x' + b (c x + d y) = a x' + b c x + b d y \text{.} \end{equation*}

Substituting \(b y = x' - a x \) from the first equation in the system gives

\begin{gather*} x'' = a x' + b c x + d (x' - a x) \\ \implies \quad x'' - (a + d) x' + (a d - b c) x = 0 \text{.} \end{gather*}

If you compute the characteristic polynomial of the general \(2 \times 2\) matrix

\begin{equation*} A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \text{,} \end{equation*}

which is also the coefficient matrix of our system of differential equations, you will find

\begin{equation*} c_A(\lambda) = \lambda^2 - (a + d) \lambda + (a d - b c) \text{,} \end{equation*}

mirroring the second-order equation we obtained from the system.

Subsection 27.3.4 Systems with complex eigenvalues

If the coefficient matrix of a coupled system of linear differential equations is not diagonalizable as a real matrix but is diagonalizable as a complex matrix, the solutions to the decoupled system will involve complex exponential functions. Complex eigenvalues always come in conjugate pairs, and using this fact along with Euler's formula (see (\(\star\)) in Subsection A.3.2) it is possible to get real solutions to the coupled system that involve combinations of cosine and sine, with frequencies related to the imaginary part of a complex eigenvalue pair and amplitudes related to the real part. But we will not pursue that discussion here — you will likely encounter this case in a first course in ordinary differential equations.