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- 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/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.06%3A_Numerical_Integration/3.6E%3A_Exercises_for_Section_3.6This page provides exercises on approximating integrals using numerical methods like the midpoint rule, trapezoidal rule, and Simpson's rule, detailing specific integrals, subdivisions, and formats fo...This page provides exercises on approximating integrals using numerical methods like the midpoint rule, trapezoidal rule, and Simpson's rule, detailing specific integrals, subdivisions, and formats for answers. It discusses the importance of numerical methods alongside the Fundamental Theorem of Calculus and includes tasks on estimating errors, arc lengths, and areas under curves. The text also features example calculations and a sample problem using coordinates to estimate land area.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.07%3A_Gauss-Seidel_Method/1.7E%3A_Exercises_for_Section_1.8This page provides exercises on the Gauss-Seidel method for solving systems of equations, detailing convergence criteria based on the coefficient matrix \(A\). It includes tasks for iterating using th...This page provides exercises on the Gauss-Seidel method for solving systems of equations, detailing convergence criteria based on the coefficient matrix \(A\). It includes tasks for iterating using the method with provided systems and initial guesses while calculating maximum absolute relative errors. The exercises conclude with approximate solutions after three iterations for various systems.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/05%3A_Introduction_to_Differential_EquationsA goal of this chapter is to develop solution techniques for different types of differential equations. As the equations become more complicated, the solution techniques also become more complicated, ...A goal of this chapter is to develop solution techniques for different types of differential equations. As the equations become more complicated, the solution techniques also become more complicated, and in fact an entire course could be dedicated to the study of these equations. In this chapter we study several types of differential equations and their corresponding methods of solution.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.04%3A_Mixing_Problems/2.4E%3A_Exercises_for_Section_2.4This page presents a collection of mixture problems about tanks and salt solutions, focusing on scenarios involving varying inflow and outflow rates. It explores how to calculate salt quantities in ta...This page presents a collection of mixture problems about tanks and salt solutions, focusing on scenarios involving varying inflow and outflow rates. It explores how to calculate salt quantities in tanks over time using differential equations and examines long-term concentration behaviors. The exercises encourage the use of numerical methods to verify predictions regarding salt concentration changes and derive specific functions for various conditions.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.10%3A_Numerical_Methods_-_Eulers_MethodThis page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-ord...This page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-order accuracy, noting the error reduction with smaller step sizes. The text also addresses challenges like numerical instability and the importance of selecting appropriate methods and step sizes.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.07%3A_Gauss-Seidel_MethodWe consider an iterative method of solving a linear system in this section. Elimination methods, such as Gaussian elimination, are prone to large round-off errors for a large set of equations. Iterati...We consider an iterative method of solving a linear system in this section. Elimination methods, such as Gaussian elimination, are prone to large round-off errors for a large set of equations. Iterative methods, such as the Gauss-Seidel method, give the user control of the round-off error.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.09%3A_Special_FactorizationsIn this section we explore two important factorizations of matrices: the Cholesky Factorization and the QR Factorization.
