Note: Remember that verifying a function is a solution to a differential equation does not require you solve the differential equation — you just need to verify that left-hand and right-hand sides evaluate to the same result when the proposed solution function is substituted.
Differentiate the expression for \(w(t)\) to get a formula for \(w'(t)\) in terms of \(x'(t)\) and \(y'(t)\text{.}\) Then substitute the expressions for \(x'(t)\) and \(y'(t)\) from the original differential equations into your expression for \(w'(t)\text{.}\) Simplify, and then see if you can relate what you have back to the change-of-variable expressions for \(w(t)\) and \(z(t)\text{.}\) Then repeat for \(z(t)\text{.}\)
If we want to convert these solutions for \(w(t), z(t)\) to solutions to for \(x(t),y(t)\text{,}\) we’ll need to reverse the change of variables. That is, we’ll need to solve the system
\begin{equation*}
\begin{sysofeqns}{rcrcr}
- x \amp + \amp y \amp = \amp w \text{,} \\
x \amp - \amp 2 y \amp = \amp z \text{,}
\end{sysofeqns}
\end{equation*}
Wait! The pattern of the equations above looks familiar … Perhaps we could use linear algebra to solve them. (And maybe use matrix inversion to solve, instead of row reducing.)
Once you’ve solved the reverse change of variables, express the solutions for \(x(t)\) and \(y(t)\) as combinations of the solutions for \(w(t)\) and \(z(t)\text{.}\)
But the original system of differential equations involving \(x(t),y(t)\) also looks linear. Can you write that differential system in terms of matrix multiplication as well? One side of your matrix equation should involve a \(2 \times 2\) matrix times \(\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right]\text{.}\) Can you convert the other “differential” side of the matrix equation into an expression involving \(\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right]\text{?}\)
Can you also turn your simplified system involving \(w(t),w'(t),z'(t),z(t)\) into a matrix equation? What do you notice about coefficient matrix in this system?
a coefficient matrix relating \(\ddt \left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \) to \(\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \text{,}\)
a coefficient matrix relating \(\ddt \left[\begin{smallmatrix} w \\ z \end{smallmatrix}\right] \) to \(\left[\begin{smallmatrix} w \\ z \end{smallmatrix}\right] \text{,}\) and
a coefficient matrix relating coordinate systems \(\left[\begin{smallmatrix} w \\ z \end{smallmatrix}\right] \) and \(\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \text{.}\)
Work out the pattern of Discovery 27.2. Suppose matrices \(A,B\) are similar via transition matrix \(P\) in the similarity relation \(\inv{P}AP = B\text{,}\) and that \(y_1(t), y_2(t), \dotsc, y_n(t)\) are functions that satisfy the differential matrix equation
Substitute the similarity relation into the differential matrix equation and rearrange to get a new differential equation
\begin{equation*}
\ddt \uvec{w}(t) = B \uvec{w}(t) \text{,}
\end{equation*}
where \(\uvec{w}(t)\) is some change of variables from \(\uvec{y}(t)\text{.}\) Be explicit about how your change of variables relates \(\uvec{y}(t)\) and \(\uvec{w}(t)\text{.}\)
Discovery 27.2 demonstrated that if a differential matrix equation \(\ddt \uvec{y}(t) = A \uvec{y}(t) \) has a diagonalizable coefficient matrix \(A\text{,}\) then a change of variables via a transition matrix \(P\) that diagonalizes \(A\) will decouple the underlying system of equations, leaving simple proportional differential equations that are solved by exponential functions \(w_j(t) = c_j e^{k_j t}\text{.}\)
What do the constants \(k_j\) represent relative to the diagonal coefficient matrix \(\inv{P} A P\text{?}\) What do they represent relative to the original coefficient matrix \(A\text{?}\)
You can obtain the first equation by combining our definitions of \(y_1(t)\) and \(y_2(t)\text{.}\) For the second equation, use the original differential equation and the fact that
