Question Number 219952 by Mamadi last updated on 04/May/25
$${solve}\:{the}\:{system}\:{differential} \\ $$$${x}'=\mathrm{3}{x}−{y}+{z} \\ $$$${y}'=\mathrm{2}{x}+{z} \\ $$$${z}'=−\mathrm{2}{x}+{y} \\ $$$${with}\:{x},{y},{and}\:{z}\:{are}\:{the}\:{function}\:{of}\: \\ $$$${t}. \\ $$$$ \\ $$
Commented by MrGaster last updated on 04/May/25
\begin{solution}
Let $\mathbf{X} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, and the system of equations is:
$$
\mathbf{X}' = \begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -2 & 1 & 0 \end{pmatrix}\mathbf{X} + \begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}
$$
\begin{enumerate}
\item \text{Solution for the homogeneous equation $\mathbf{X}' = A\mathbf{X}$:}
$$
\det(A - \lambda I) = -\lambda^3 + 3\lambda^2 - 4\lambda + 2 = 0 \Rightarrow (\lambda - 1)(\lambda^2 - 2\lambda + 2) = 0
$$
\text{Eigenvalues:}
$$
\lambda_1 = 1, \quad \lambda_2 = 1 + i, \quad \lambda_3 = 1 - i
$$
\item \text{Corresponding eigenvectors and homogeneous solutions:}
\begin{itemize}
\item \text{For $\lambda_1 = 1$, eigenvector $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$, solution:}
$$
\mathbf{X}_1 = C_1 e^{t} \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}
$$
\item \text{For $\lambda_2 = 1 + i$, eigenvector $\mathbf{v}_2 = \begin{pmatrix} 1 + i \\ 2 \\ -1 - i \end{pmatrix}$, real and imaginary solutions:}
$$
\mathbf{X}_2 = e^{t} \left[ C_2 \begin{pmatrix} \cos t - \sin t \\ 2\cos t \\ -\cos t + \sin t \end{pmatrix} + C_3 \begin{pmatrix} \sin t + \cos t \\ 2\sin t \\ -\sin t - \cos t \end{pmatrix} \right]
$$
\end{itemize}
\item \text{Particular solution for the non-homogeneous system:}
\text{Assume a constant solution $\mathbf{X}_p = \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}$, verified to satisfy:}
$$
A\mathbf{X}_p + \mathbf{B} = \mathbf{0}
$$
\item \text{General solution:}
$$
\mathbf{X} = \mathbf{X}_h + \mathbf{X}_p
$$
\text{Explicitly:}
$$
\begin{aligned}
x(t) &= C_1 e^{t} + e^{t} \left( (C_2 + C_3)\cos t + (C_3 - C_2)\sin t \right) - 1, \\
y(t) &= 2C_1 e^{t} + 2e^{t} \left( C_2 \cos t + C_3 \sin t \right) - 2, \\
z(t) &= e^{t} \left( (-C_2 - C_3)\cos t + (C_2 - C_3)\sin t \right) + 1.
\end{aligned}
$$
\end{enumerate}
\boxed{
\begin{aligned}
x(t) &= C_1 e^{t} + e^{t} \left( (C_2 + C_3)\cos t + (C_3 - C_2)\sin t \right) - 1, \\
y(t) &= 2C_1 e^{t} + 2e^{t} \left( C_2 \cos t + C_3 \sin t \right) - 2, \\
z(t) &= e^{t} \left( (-C_2 - C_3)\cos t + (C_2 - C_3)\sin t \right) + 1.
\end{aligned}
}
\end{solution}
Commented by mr W last updated on 04/May/25
$${can}\:{you}\:{please}\:\:{post}\:{your}\:{answer}\:{in} \\ $$$${proper}\:{readable}\:{format}? \\ $$
Commented by mr W last updated on 04/May/25
Commented by MrGaster last updated on 04/May/25
