Linear Algebra and Differential Equations •
Math 2250
At Utah State University, Math 2250 covers linear algebra and differential equations. This course is primarily oriented towards students in engineering. This page provides a summary of most topics covered in Linear Algebra and Differential Equations . This is a list that summarizes derivative, integration and identity rules.
Differential Equations
As Differential Equations are a new topic to students learning the material for the first time, the terminology can be overwhelming and confusing. Important terms to know and understand are the following:
- Differential Equation - An expression that relates a function to its derivative
- Linear - Sequential and proportional, in math, linearity relates to being able to operate on every part of an equation independently.
- Homogeneous - Consisting of a uniform type of function. The differential equation only consists of a function and its derivatives.
- Non-homogeneous - A differential equation with a forcing term.
Differential equations are used to model engineering, finance, and ecological problems.
First Order Differential Equations
The general form of a first order differential equation is shown below.
\[y' - a y = f(t)\]
The \(y'\) term is the first derivative of a function \(y\). \(f(t)\) is the forcing term or forcing function. The \(ay\) term is the function multiplied by a rate. If the differential equation given has the forcing term \(f(t)=0\), then the equation is homogeneous.
This section on First Order Differential Equations covers the following topics:
- Homogenous Differential Equations
- Non-Homogenous Differential Equations with Constant, Exponential, Sinusoidal, Step, and Delta Forcing Terms
- Solutions to Any Input with Integrating Factor and Green's Function
- Logistic Equations with Substitution and Separation of Variables
- General Solutions
Review First Order Differential Equations
Second Order Differential Equations
Second-order differential equations relate a function to its first and second derivatives. A general second-order differential equation is usually written as \[my''+\gamma y' +ky = f(t)\]
This section on Second Order Differential Equations covers the following topics
- Homogeneous Second Order Equations with Characteristic Polunomial, Undamped Harmonic Oscillator, General Solution Behavior, Under-damped System,and Over-damped System
- Non-Homogenous Second-Order Equations with Method of Undetermined Coefficients for Exponentials, Sinusoids, and Polynomials.
- Green's Function
Review Second Order Differential Equations
Complex and Resonant Solutions for First and Second Order Equations
Complex Solutions utilize Euler' formula, written below, and Resonant solutions utilize the impulse response for a differential equation which can be found in the explanation of Green's Function.\[Re^{i\omega t}=R\cos(\omega t)+iR\sin(\omega t)\]
This section covers the following topics:
- Euler's Formula and Complex Roots
- Deriving Trigonometric Identities using Euler's Formula
- Solutions to Differential Equations using Euler's Formula
- Resonance Solutions
Review Complex and Resonant Solutions
Laplace Transforms and Convolution
Laplace Transforms are a powerful tool for solving differential equations. A Laplace transform is an operator that can be applied to an equation that transforms the function domain from time to Laplace space. This transform allows for any differential equation to be solved algebraically. The method of transforming a function into Laplace space is the following.
\[\mathcal{L}[f(t)]=\int_0^\infty e^{-st}f(t)dt\]
The laplace transforms of the following common functions are covered in this section:
- Exponentials
- Sinusoids
- Polynomial
- Delta Function
- Arbitrary Functions
Solving differential equations by Laplace transforms, general laplace transforms, and the convolution function are also discussed.
Review Laplace Transforms and Convolution
Linear Algebra
Linear algebra is a powerful tool for solving systems of equations and analyzing data. In science and engineering, understanding how to create and operate matrices can allow for optimization in system design. This section introduces the basic functions of Linear Algebra and Matrix operations. Matrices are arrays of vectors in a space.
Matrix Operations
Matrix operations including matrix multiplication, shown below, are fundamental to understanding linear algebra and its applications.
$$
\begin{bmatrix}
a_1 & a_2 & \cdots & a_n
\end{bmatrix}
\times
\begin{bmatrix}
b_1\\
b_2\\
\vdots\\
b_3
\end{bmatrix}
=[a_1 \cdot b_1 + a_2 \cdot b_2 +\cdots+ a_n \cdot b_n]
$$
Matrix Operations cover the following topics:
- Eigenvalues and Matrix Stability
- Diagonalization of a Matrix
- Solving Linear Systems with Eigenvalues
- Stability of Coupled Linear Systems
Matrix Properties
Matrix Properties including finding a determinant as shown below, cover the following topics:
$$
|A| = | \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}| = |\begin{bmatrix}
a & 0\\
0 & d
\end{bmatrix}|
-|\begin{bmatrix}
b & 0\\
0 & c
\end{bmatrix}| = ad-bc
$$
- Vector Sub-spaces
- Singular Matrices
- Determinant of a Matrix
Eigenvalues and Eigenvectors
Eigenvectors are specific vectors that can be derived from a matrix, each with associated eigenvalues that scale the length of the vector. Eigenvalues for a matrix can give information about the stability of the linear system. The following expression can be used to derive eigenvalues for any square matrix.
$$
det(A-\lambda I) =
\begin{bmatrix}
n_0 & \cdots & n_f\\
\cdots & \cdots & \cdots\\
m_0 & \cdots & m_f
\end{bmatrix}
-
\lambda I = 0
$$
Eigenvalues and Eigenvectors cover the following topics:
- Vector Sub-spaces
- Singular Matrices
- Determinant of a Matrix
Review Eigenvalues and Eigenvectors
Applied Mathematics
Coming Soon