Solve the Linear Dynamical System $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =A\mathbf{x}$ by Diagonalization

Differential Equations Problems and Solutions

Problem 667

(a) Find all solutions of the linear dynamical system
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}\mathbf{x},\] where $\mathbf{x}(t)=\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ is a function of the variable $t$.

(b) Solve the linear dynamical system
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}\mathbf{x}\] with the initial value $\mathbf{x}(0)=\begin{bmatrix}
1 \\
3
\end{bmatrix}$.

 
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Solution.

(a) Find all solutions of the linear dynamical system $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}\mathbf{x}$

Note that the given system is
\[\begin{bmatrix}
\frac{\mathrm{d}{x_1}}{\mathrm{d} t} \\[6pt] \frac{\mathrm{d}{x_2}}
{\mathrm{d} t}
\end{bmatrix}
=\begin{bmatrix}
x_1 \\
3x_2
\end{bmatrix}.\] Thus, we have two uncoupled differential equations
\begin{align*}
\frac{\mathrm{d}{x_1}}{\mathrm{d} t} &=x_1 \\[6pt] \frac{\mathrm{d}{x_2}}{\mathrm{d} t} &=3x_2.
\end{align*}

The solutions to these differential equations are
\begin{align*}
x_1(t)&=e^t x_1(0)\\
x_2(t)&=e
^{3t} x_2(0).
\end{align*}
Thus we see that
\[\mathbf{x}(t)=\begin{bmatrix}
e^t x_1(0) \\[6pt] e
^{3t} x_2(0)
\end{bmatrix}.\]

(b) Solve the linear dynamical system $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}\mathbf{x}$

Let $A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}$.
Then the matrix $A$ has eigenvalues $1, 3$ and corresponding eigenvectors are
\[\begin{bmatrix}
1 \\
1
\end{bmatrix} \text{ and } \begin{bmatrix}
-1 \\
1
\end{bmatrix},\] respectively.
(See the post Diagonalize a 2 by 2 Symmetric Matrix for details.)

Thus, if we put $S=\begin{bmatrix}
1 & -1\\
1& 1
\end{bmatrix}$, then $A$ is diagonalizable by $S$. That is,
\[S^{-1}AS=D,\] where
\[D=\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}.\]


Substituting $A=SDS^{-1}$ into the given system, we have
\begin{align*}
\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}&=(SDS^{-1})\mathbf{x}\\[6pt] \Leftrightarrow \quad S^{-1}\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}&=(DS^{-1})\mathbf{x}\\[6pt] \Leftrightarrow \quad \frac{\mathrm{d}(S^{-1} \mathbf{x})}{\mathrm{d}t}&=(DS^{-1})\mathbf{x}.
\end{align*}

Let $\mathbf{u}(t)=S^{-1}\mathbf{x}$. Then we obtain the system
\[\frac{\mathrm{d}\mathbf{u}}{\mathrm{d}t} =\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}\mathbf{u}.\] We solved this system in part (a) and the general solution is given by
\[\mathbf{u}(t)=\begin{bmatrix}
e^t u_1(0) \\[6pt] e^{3t} u_2(0)
\end{bmatrix}.\]

We determine the values of $u_1(0)$ and $u_2(0)$ using the given initial value $\mathbf{x}(0)=\begin{bmatrix}
1 \\
3
\end{bmatrix}$.
We have
\[\begin{bmatrix}
u_1(0) \\
u_2(0)
\end{bmatrix}=\mathbf{u}(0)=S^{-1} \mathbf{x}(0)=\frac{1}{2}\begin{bmatrix}
1 & 1\\
-1& 1
\end{bmatrix}\begin{bmatrix}
1 \\
3
\end{bmatrix}=\begin{bmatrix}
2 \\
1
\end{bmatrix}.\] (Note that $\begin{bmatrix}
2 \\
1
\end{bmatrix}$ is the coordinate vector of $\mathbf{x}(0)=\begin{bmatrix}
1 \\
3
\end{bmatrix}$ with respect to the eigenbasis $\begin{bmatrix}
1 \\
1
\end{bmatrix}, \begin{bmatrix}
-1 \\
1
\end{bmatrix}$.)

It follows that the solution of the original system is
\begin{align*}
\mathbf{x}=S\mathbf{u}=\begin{bmatrix}
1 & -1\\
1& 1
\end{bmatrix}
\begin{bmatrix}
2e^t \\[6pt] e^{3t}
\end{bmatrix}
=2e^t\begin{bmatrix}
1 \\
1
\end{bmatrix}+ e^{3t}\begin{bmatrix}
-1 \\
1
\end{bmatrix}.
\end{align*}

Another Solution of (b)

In this solution, we use the following theorem.

Theorem. Let $A$ be a diagonalizable $n\times n$ matrix.
Let $\{\mathbf{v}_1,\dots, \mathbf{v}_n\}$ be an eigenbasis for $A$, with associated eigenvalues $\lambda_1, \dots, \lambda_n$. Then the general solution of the linear dynamical system
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =A\mathbf{x}\] is
\[\mathbf{x}(t)=c_1 e^{\lambda_1 t}\mathbf{v}_1+\cdots +c_n e^{\lambda_n t}\mathbf{v}_n,\] where $c_1, \dots, c_n$ are arbitrary complex numbers.

As in the above solution, we know that the matrix $A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}$ has eigenvalues $1, 3$ and corresponding eigenvectors are
\[\begin{bmatrix}
1 \\
1
\end{bmatrix} \text{ and } \begin{bmatrix}
-1 \\
1
\end{bmatrix},\] respectively.

So the formula in the theorem yields the general solution
\[\mathbf{x}(t)=c_1 e^{t}\begin{bmatrix}
1 \\
1
\end{bmatrix}+c_2 e^{3t}\begin{bmatrix}
-1 \\
1
\end{bmatrix},\] where $c_1, c_2$ are constants.


Since the initial is $\mathbf{x}(0)=\begin{bmatrix}
1 \\
3
\end{bmatrix}$, we have
\begin{align*}
\begin{bmatrix}
1 \\
3
\end{bmatrix}=c_1\begin{bmatrix}
1 \\
1
\end{bmatrix}+c_2\begin{bmatrix}
-1 \\
1
\end{bmatrix}.
\end{align*}
Solving this system, we obtain $c_1=2$ and $c_2=1$.
Thus, the solution of the linear dynamical system with the given initial value is
\[\mathbf{x}(t)=2 e^{t}\begin{bmatrix}
1 \\
1
\end{bmatrix}+e^{3t}\begin{bmatrix}
-1 \\
1
\end{bmatrix}.\]


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