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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
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Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Section 12: Nonhomogeneous Linear Systems
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
Links & Resources
Return to Differential Equations Page
Return to Mini-Course Main Page
Watch Concepts Video 1
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find the general solution of the nonhomogeneous system.
\[
{\bf x}'=\begin{pmatrix}
-4&2\\
2&-1\\
\end{pmatrix}{\bf x}+\begin{pmatrix}
t^{-1}\\2t^{-1}+4
\end{pmatrix}, \hskip.3in t>0\]
Show Answer
\[\textbf{x}(t)=\left(c_1+\ln(t)+\dfrac{8}{5}t\right)\begin{pmatrix}1\\2\end{pmatrix}+\left(c_2e^{-5t}+\dfrac{4}{25}\right)\begin{pmatrix}-2\\1\end{pmatrix}\]
Watch Video Solution
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V79
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.
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?
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