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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_Theory/6.4E%3A_Exercises_for_Section_6.4This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solvin...This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solving first-order systems through matrix exponentiation. Each exercise outlines problem setups, provides hints, and sometimes offers detailed solution computations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.02%3A_Gaussian_EliminationThe work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions, by representing a linear system as an...The work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions, by representing a linear system as an augmented matrix and performing a systematic set of operations on it to reach a solution to the system.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.02%3A_Classification_of_Differential_EquationsThis page discusses the classification of differential equations as ordinary or partial based on independent variables, emphasizing the importance of order and linearity. It covers concepts of homogen...This page discusses the classification of differential equations as ordinary or partial based on independent variables, emphasizing the importance of order and linearity. It covers concepts of homogeneous vs. nonhomogeneous equations, constant vs. variable coefficients, and introduces autonomous equations. Additionally, it mentions a 7-minute video providing context for understanding these concepts and solutions.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_AlgebraThis page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to e...This page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to express solutions in parametric form. The content is organized into sections that build foundational knowledge on linear equations, algorithms for solutions, and solution representation.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.06%3A_Reduction_of_Order/3.6E%3A_Exercises_for_Section_3.6This page consists of exercises aimed at finding general solutions to differential equations, using known solutions alongside specified equations. It includes examples with exponential and polynomial ...This page consists of exercises aimed at finding general solutions to differential equations, using known solutions alongside specified equations. It includes examples with exponential and polynomial combinations while deriving fundamental solution sets.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.05%3A_Higher_order_linear_ODEs/3.5E%3A_Exercises_for_Section_3.5This page offers exercises aimed at solving differential equations and understanding linear independence of functions. Tasks involve finding general solutions, determining characteristic equations fro...This page offers exercises aimed at solving differential equations and understanding linear independence of functions. Tasks involve finding general solutions, determining characteristic equations from roots, verifying function independence, and constructing equations for known solutions. The exercises are designed to strengthen comprehension of differential equations and linear algebra concepts.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.04%3A_Constant_coefficient_second_order_linear_ODEsThis page explains solving second-order linear homogeneous differential equations using the characteristic equation method, covering distinct, repeated, and complex roots. It includes the general solu...This page explains solving second-order linear homogeneous differential equations using the characteristic equation method, covering distinct, repeated, and complex roots. It includes the general solution forms and adaptations for double roots, along with examples. Additionally, it discusses complex numbers, their properties, and Euler's formula, applying these concepts to derive trigonometric identities.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.07%3A_Variation_of_Parameters/3.7E%3A_Exercises_for_Section_3.7This page outlines exercises focused on solving differential equations through the method of variation of parameters. It includes various equations, some with solutions, and tasks that involve applyin...This page outlines exercises focused on solving differential equations through the method of variation of parameters. It includes various equations, some with solutions, and tasks that involve applying this method to linear equations and initial value problems. Key mathematical principles, such as Abel's formula for Wronskian, are emphasized.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.02%3A_Linear_systems_of_ODEs/6.2E%3A_Exercises_for_Section_6.2This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve find...This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve finding general solutions, validating specific solutions for given matrices, and deriving forms with mathematical justification. The exercises explore various systems and functions, necessitating in-depth understanding and presentation of results linked to independence and matrix notation.
- https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/10%3A_Signed_Numbers/10.09%3A_Proficiency_ExamExercise \PageIndex1 Write all integers that are strictly between –8 and –3. Exercise \PageIndex2 Write all integers that are between and including –2 and 1. For problems 3-5, write the ap...Exercise \PageIndex1 Write all integers that are strictly between –8 and –3. Exercise \PageIndex2 Write all integers that are between and including –2 and 1. For problems 3-5, write the appropriate symbol (<, >, =) in place of the □ for each pair of numbers. Exercise \PageIndex3 Exercise \PageIndex4 Exercise \PageIndex5 For problems 6 and 7, what numbers can replace x so that the statement is true? −4≤x≤0, x is a natural number.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.02%3A_The_Method_of_Undetermined_Coefficients_IThis section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special ...This section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special form that is still sufficiently general to occur in many applications. This sections makes extensive use of the idea of variation of parameters introduced previously.