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  • https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.10%3A_Numerical_Methods_-_Eulers_Method
    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-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/Bookshelves/Calculus/Differential_Calculus_for_the_Life_Sciences_(Edelstein-Keshet)/12%3A_Solving_Differential_Equations/12.03%3A_Eulers_Method_and_Numerical_Solutions
    We sometimes need a method for computing an approximation for the desired solution - referred to as a "numerical solution". The idea is to harness a computational device to find numerical values of po...We sometimes need a method for computing an approximation for the desired solution - referred to as a "numerical solution". The idea is to harness a computational device to find numerical values of points along the solution curve, rather than attempting to determine the formula for the solution as a function of time. We illustrate this process using a technique called Euler’s method, which is based on an approximation of a derivative by the slope of a secant line.
  • https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/05%3A_Modular_Arithmetic_and_Primes/5.02%3A_New_Page
    A reduced set of residues R modulo b is a set φ(b) integers in Z, such R has exactly one integer in each class congruent to p{1,b1} (modulo \(b\...A reduced set of residues R modulo b is a set φ(b) integers in Z, such R has exactly one integer in each class congruent to p{1,b1} (modulo b) such that p is relatively prime to b. If the numbers {xi} form a complete set of residues modulo b (a reduced set of residues modulo b), then {axi} is a complete set of residues modulo b (a reduced set of residues modulo b).

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