Definition 4.3.1. Fundamental matrix.
An \(n \times n\) matrix-valued function \(\Phi\) is called a fundamental matrix to (4.1.1) if \(\Phi'(t)=A\Phi(t)\) and for each \(t\text{,}\) \(\Phi(t)\) is an invertible matrix.
Section 4.3 Variation of parameters
Definition 4.3.2. Nonohomogeneous systems.
A nonohomogeneous vector equation is of the form
\begin{equation}
\vec{x}'(t)=A\vec{x}(t)+\vec{u}(t),\tag{4.3.1}
\end{equation}
where \(\vec{x}\) is the state vector and \(\vec{u}\) is the control vector.
Theorem 4.3.3. Variation of parameters for systems.
The solution to (4.3.1) is of the form \(\vec{x}=\vec{x}_h+\vec{x}_p\text{,}\) where \(\vec{x}_h(t)=\Phi(t)\vec{c}\) is the homogeneous solution for some constant vector \(\vec{c}\) and \(\vec{x}_p=\Phi(t)\displaystyle\int \Phi^{-1}(t)\vec{u}(t)\mathrm{d}t\) is the particular solution.
Proof.
Example 4.3.4.
Use variation of parameters to find a general solution to the nonohomogeneous system
\begin{equation*}
\vec{x}'(t)=\begin{bmatrix} 5& -1 \\ 3& 1 \end{bmatrix} \vec{x}(t) + \begin{bmatrix} 2 \\ 1 \end{bmatrix}e^{2t},
\end{equation*}
where \(\Phi(t)=\begin{bmatrix} e^{2t} & e^{4t} \\ 3e^t & e^{4t} \end{bmatrix}\) is a fundamental matrix of the associated homogeneous system on \((-\infty,\infty)\text{.}\)
Example 4.3.5.
Use variation of parameters to find a general solution to the nonohomogeneous system
\begin{equation*}
\vec{x}'(t)=\begin{bmatrix} 1& -1 \\ 1& -1 \end{bmatrix} \vec{x}(t) + \begin{bmatrix} 1 \\ 1 \end{bmatrix}t^{-1},
\end{equation*}
where \(\Phi(t)=\begin{bmatrix} t+1 & 1 \\ t & 1 \end{bmatrix}\) is a fundamental matrix of the associated homogeneous system on \((0,\infty)\text{.}\)
Example 4.3.6.
Use variation of parameters to find a general solution to the nonohomogeneous system
\begin{equation*}
\vec{x}'(t)=\begin{bmatrix} 0& 2 \\ -2& 0 \end{bmatrix} \vec{x}(t) + \begin{bmatrix} 0 \\ 2 \end{bmatrix}t^{-1},
\end{equation*}
where \(\Phi(t)=\begin{bmatrix} \cos(2t) & \sin(2t) \\ -\sin(2t) & \cos(2t) \end{bmatrix}\) is a fundamental matrix of the associated homogeneous system.
