# Section 12: Nonhomogeneous Linear Systems

### Instructions

• First, you should watch the concepts videos below explaining the topics in the section.
• Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
• Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
• This is the last section so you should return to the main differential equations page when you have finished the material.

### Concepts

• Solving a nonhomogeneous system of equations
• Using the method of Variation of Parameters (Lagrange method) to find a particular solution

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Find the general solution of the nonhomogeneous system.
${\bf x}'=\begin{pmatrix} -4&2\\ 2&-1\\ \end{pmatrix}{\bf x}+\begin{pmatrix} t^{-1}\\2t^{-1}+4 \end{pmatrix}, \hskip.3in t>0$

$\textbf{x}(t)=\left(c_1+\ln(t)+\dfrac{8}{5}t\right)\begin{pmatrix}1\\2\end{pmatrix}+\left(c_2e^{-5t}+\dfrac{4}{25}\right)\begin{pmatrix}-2\\1\end{pmatrix}$

To see the full video page and find related videos, click the following link.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Explain the Lagrange method (Variation of Parameters) for solving a non homogeneous linear system of ODEs. What is the most important ingredient of the formulas?