Find the general solution of the system.
\begin{equation*}
\begin{cases} x' \amp = 2x+3y+5 \\ y' \amp = 2x+y-2t \end{cases}
\end{equation*}
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Handout Lesson 30, Nonhomogeneous Linear Systems of Differential Equations
Textbook Section(s).
This lesson is based on Section 5.7 of your textbook by Edwards, Penney, and Calvis.
The Method of Undetermined Coefficients.
We used a method of undetermined coefficients to find particular solutions for a single non-homogeneous linear differential equation with constant coefficients. This method can be expanded to solving a nonhomogeneous linear system of differential equations with constant coefficients:
\begin{gather}
\mathbf{x'}=\mathbf{A}\mathbf{x} +\mathbf{f}(t)\tag{#}
\end{gather}
where \(\mathbf{A}\) is an \(n\times n\) matrix of constants. The procedure is a generalization of the procedure we used when working with a single equation. We will demonstrate the procedure with two examples.
Example 193. Method of undetermined coefficients.
Example 194. Guesses for the method of undetermined coefficients.
(Based on number 5 from Section 5.7 of your textbook by Edwards, et.al.)
What would you try for \(\mathbf{x}_p\) if you were using the method of undetermined coefficients to solve the following system?
\begin{equation*}
\mathbf{x'}= \left[ \begin{array}{cc} 6 \amp -7 \\ 1 \amp -2
\end{array} \right] \mathbf{x}+ \left[ \begin{array}{c} 10 \\
-2e^{-t} \end{array} \right]
\end{equation*}
What lesson were you supposed to learn from this example? How is the method of undetermined coefficients different for systems than it is for a single equation?
Variation of Parameters.
The method of variation of parameters for linear systems of differential equations is easier to describe when we write the system using matrix notation. Consider the nonhomogeneous linear system of differential equations in which the coefficients do not need to be constants:
\begin{gather}
\mathbf{x'}=\mathbf{P}(t)\mathbf{x} +\mathbf{f}(t)\tag{✶}
\end{gather}
where \(\mathbf{P}(t)\) is an \(n\times n\) matrix of functions. The associated homogeneous system is
\begin{gather}
\mathbf{x'}=\mathbf{P}(t)\mathbf{x}\tag{✶✶}
\end{gather}
Let \(\mathbf{\Phi}(t)\) be a fundamental matrix for the system in (✶✶). Then each column of \(\mathbf{\Phi(t)}\) is a solution to (✶✶). Thus
\begin{gather}
\mathbf{\Phi'}(t)=\mathbf{P}(t)\mathbf{\Phi}(t)\tag{✶✶✶}
\end{gather}
Because the columns of \(\mathbf{\Phi}\) are linearly independent, any solution of (✶✶) has the form
\begin{equation*}
\mathbf{x}_c(t)=\underline{\hspace{3in}}
\end{equation*}
We want to find a solution of (✶) of the form
\begin{equation*}
\mathbf{x}_p(t) = \underline{\hspace{3in}}
\end{equation*}
Substituting \(\mathbf{x}_p(t)\) into (✶) yields:
Theorem 195. Variation of Parameters.
If the nonhomogeneous system of linear equations
\begin{gather}
\mathbf{x'}=\mathbf{P}(t)\mathbf{x} +\mathbf{f}(t)\tag{†}
\end{gather}
has fundamental matrix \(\mathbf{\Phi}(t)\) on some interval \(I\) where \(\mathbf{P}(t)\) and \(\mathbf{f}(t)\) are continuous, then (†) has a particular solution of the form
\begin{equation*}
\mathbf{x_p}(t)=\mathbf{\Phi}(t)\mathbf{u}(t)
\end{equation*}
where the derivative of \(\mathbf{u}(t)\) satisfies
\begin{equation*}
\mathbf{u'}(t)=\mathbf{\Phi^{-1}(t)}\mathbf{f}(t)
\end{equation*}
Example 196. Variation of parameters.
(Number 21 from Section 9.7 in the Differential Equations textbook by Nagle, Saff, and Snider)
Use variation of parameters to solve the initial value problem.
\begin{equation*}
\mathbf{x'}(t)= \left[ \begin{array}{cc} 0 \amp 2 \\ -1 \amp 3
\end{array} \right] \mathbf{x}(t)+ \left[ \begin{array}{c} e^t
\\ -e^{t} \end{array} \right] \qquad \mathbf{x}(0)= \left[
\begin{array}{c} 5 \\ 4 \end{array} \right]
\end{equation*}
Homework Comments.
In your homework, they ask you to use variation of parameters to solve a nonhomogeneous system of linear equations with constant coefficients:
\begin{equation*}
\mathbf{x'(t)}=\mathbf{A}\mathbf{x(t)}+\mathbf{f}(t), \qquad
\mathbf{x}(a)=\mathbf{x_a}
\end{equation*}
They also give you \(e^{\mathbf{A}t}\text{.}\) This is your \(\mathbf{\Phi}(t)\text{.}\) In this case,
\begin{gather}
\mathbf{\Phi}^{-1}(t)= \underline{\hspace{5in}}\tag{#}
\end{gather}
In your written homework, they ask you to “use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problem.” You are asked to do number 25. I strongly suggest that you use some CAS (computer algebra system) to help you solve this problem. The computations are a little intense. In Brightspace, I have posted some notes that come with your textbook if you want to use Maple, Mathematica, or MATLAB. I used Sagemath, which is available for free at
https://cocalc.com/features/sage. In Brightspace, I posted my Sage worksheet for solving Problem 24 from Section 3.7 with
\begin{equation*}
\mathbf{A}= \left[
\begin{array}{rr} 3 \amp -1 \\
9 \amp -3
\end{array} \right] \quad
\mathbf{f}(t)= \left[
\begin{array}{c}
0 \\ t^{-2}
\end{array} \right] \quad
\mathbf{x}(1)= \left[
\begin{array}{c}
3 \\ 7
\end{array} \right] \quad
e^{\mathbf{A}t}= \left[ \begin{array}{cc}
1+3t \amp -t \\
9t \amp 1-3t
\end{array}
\right]
\end{equation*}
