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5.3: Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/05%3A_Linear_Second_Order_Equations/5.03%3A_Constant_Coefficient_Homogeneous_Equations
Since ${-b\over a}$ is just a constant we will replace it with the constant $r$ and we get a solution of the form $y=e^{rx}\nonumber$ Since $r=-1$ and $r=-5$ are roots, $y_1=e^{-x}$ and \(...Since ${-b\over a}$ is just a constant we will replace it with the constant $r$ and we get a solution of the form $y=e^{rx}\nonumber$ Since $r=-1$ and $r=-5$ are roots, $y_1=e^{-x}$ and $y_2=e^{-5x}$ are solutions of Equation $\ref{eq:5.2.4}$ and the general solution of Equation $\ref{eq:5.2.4}$ is $y_1=e^{(\lambda + \omega i)x}=e^{\lambda x}e^{i\omega x}=e^{\lambda x}(\cos \omega x+i\sin \omega x)=e^{\lambda x}\cos \omega x+ie^{\lambda x}\sin \omega x)\nonumber$
12.2: The Eigenvalue-Eigenvector Equation
https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/12%3A_Eigenvalues_and_Eigenvectors/12.02%3A_The_Eigenvalue-Eigenvector_Equation
The left hand side of this equation is a polynomial in the variable $\lambda$ called the $\textit{characteristic polynomial}$ $P_{M}(\lambda)$ of $M$ . \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\l...The left hand side of this equation is a polynomial in the variable $\lambda$ called the $\textit{characteristic polynomial}$ $P_{M}(\lambda)$ of $M$ . $P_{M}(\lambda)=(\lambda-\lambda_{1})(\lambda-\lambda_{2})\cdots(\lambda-\lambda_{n})\: \Longrightarrow\: P_{M}(\lambda_{i})=0$ An obvious candidate is the exponential function, $e^{\lambda x}$ ; indeed, $\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}$ .
4.3: Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/04%3A_Linear_Second_Order_Equations/4.03%3A_Constant_Coefficient_Homogeneous_Equations
This section deals with homogeneous equations of the special form $ay''+by'+cy=0,$ where $a$ , $b$ , and $c$ are constant ( $a\ne0$ ). When you've completed this section you'll know everything...This section deals with homogeneous equations of the special form $ay''+by'+cy=0,$ where $a$ , $b$ , and $c$ are constant ( $a\ne0$ ). When you've completed this section you'll know everything there is to know about solving such equations.
4.4: Higher Order Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/04%3A_Linear_Second_Order_Equations/4.04%3A_Higher_Order_Constant_Coefficient_Homogeneous_Equations
In this section we consider the homogeneous constant coefficient equation of n-th order.
4.2: Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/04%3A_Linear_Second_Order_Equations/4.02%3A_Constant_Coefficient_Homogeneous_Equations
This section deals with homogeneous equations of the special form $ay''+by'+cy=0,$ where $a$ , $b$ , and $c$ are constant ( $a\ne0$ ). When you've completed this section you'll know everything...This section deals with homogeneous equations of the special form $ay''+by'+cy=0,$ where $a$ , $b$ , and $c$ are constant ( $a\ne0$ ). When you've completed this section you'll know everything there is to know about solving such equations.
5.4: Diagonalization
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.03%3A_Diagonalization
This page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as $A = CDC^{-1}$ with $D$ diagonal. It discusses the Diagonalization Theorem, eig...This page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as $A = CDC^{-1}$ with $D$ diagonal. It discusses the Diagonalization Theorem, eigenspaces, eigenvalues, and the significance of linear independence among eigenvectors. Multiple diagonal forms can arise, while geometric and algebraic multiplicities influence diagonalizability.
8.5.1: The Eigenvalue-Eigenvector Equation
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/08%3A_Spectral_Theory/8.05%3A_Supplemental_Notes_-_More_on_Eigenvalues_and__Intro_to_Eigenspaces/8.5.01%3A_The_Eigenvalue-Eigenvector_Equation
The left hand side of this equation is a polynomial in the variable $\lambda$ called the $\textit{characteristic polynomial}$ $P_{M}(\lambda)$ of $M$ . \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\l...The left hand side of this equation is a polynomial in the variable $\lambda$ called the $\textit{characteristic polynomial}$ $P_{M}(\lambda)$ of $M$ . $P_{M}(\lambda)=(\lambda-\lambda_{1})(\lambda-\lambda_{2})\cdots(\lambda-\lambda_{n})\: \Longrightarrow\: P_{M}(\lambda_{i})=0$ An obvious candidate is the exponential function, $e^{\lambda x}$ ; indeed, $\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}$ .
3.3: Diagonalization and Eigenvalues
https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/03%3A_Determinants_and_Diagonalization/3.03%3A_Diagonalization_and_Eigenvalues
The world is filled with examples of systems that evolve in time—the weather in a region, the economy of a nation, the diversity of an ecosystem, etc. Describing such systems is difficult in general a...The world is filled with examples of systems that evolve in time—the weather in a region, the economy of a nation, the diversity of an ecosystem, etc. Describing such systems is difficult in general and various methods have been developed in special cases. In this section we describe one such method, called diagonalization, which is one of the most important techniques in linear algebra.
6.2: The Characteristic Polynomial
https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/06%3A_Eigenvalues_and_Eigenvectors/6.02%3A_The_Characteristic_Polynomial
In Section 1 we discussed how to decide whether a given number λ is an eigenvalue of a matrix, and if so, how to find all of the associated eigenvectors. In this section, we will give a method for c...In Section 1 we discussed how to decide whether a given number λ is an eigenvalue of a matrix, and if so, how to find all of the associated eigenvectors. In this section, we will give a method for computing all of the eigenvalues of a matrix. This does not reduce to solving a system of linear equations: indeed, it requires solving a nonlinear equation in one variable, namely, finding the roots of the characteristic polynomial.
6.3E: Exercises for Section 6.3
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.03%3A_Diagonalization/6.3E%3A_Exercises_for_Section_6.3
This page presents exercises on finding eigenvalues and eigenvectors for matrices, assessing diagonalizability, and applying the Cayley-Hamilton theorem. Each exercise includes matrices, known eigenva...This page presents exercises on finding eigenvalues and eigenvectors for matrices, assessing diagonalizability, and applying the Cayley-Hamilton theorem. Each exercise includes matrices, known eigenvalues, eigenvectors, and diagonalizability status, along with hints for dealing with complex eigenvalues and deriving the characteristic polynomial. The objective is to help students understand diagonalization and matrix theory through guided challenges and proofs.
9.2: Higher Order Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/09%3A_Linear_Higher_Order_Differential_Equations/9.02%3A_Higher_Order_Constant_Coefficient_Homogeneous_Equations
In this section we consider the homogeneous constant coefficient equation of n-th order.

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