Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/2%3A_First_Order_Differential_Equations/2.8%3A_Theory_of_Existence_and_UniquenessIf a first order differential satisfies continuity conditions, then the initial value problem will have a unique solution in some neighborhood of the initial value.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/03%3A_Functions/3.05%3A_Proof_by_ContradictionThis page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain ...This page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain element maps to a codomain element, and using contradiction to establish uniqueness in mapping. It also introduces mapping composition and explains its operation with two mappings.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.07%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_EquationsAlthough there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximati...Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.04%3A_Slope_Fields/1.4E%3A_Exercises_for_Section_1.4This page contains exercises focused on differential equations and slope fields, including sketching slope fields, analyzing behaviors, and exploring the existence and uniqueness of solutions. It emph...This page contains exercises focused on differential equations and slope fields, including sketching slope fields, analyzing behaviors, and exploring the existence and uniqueness of solutions. It emphasizes continuity and differentiability in relation to Picard's theorem and tasks students with matching equations to slope fields and deducing solutions from specific values.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.01%3A_Matrix_Operations/2.1E%3A_Exercises_for_Section_2.1This page presents various exercises on matrix operations, including addition, multiplication, conditions for operation definitions, and properties of matrices. It covers topics like commuting matrice...This page presents various exercises on matrix operations, including addition, multiplication, conditions for operation definitions, and properties of matrices. It covers topics like commuting matrices, zero products, and the distinction between symmetric and skew-symmetric matrices. Key exercises involve calculating specific matrix entries, demonstrating the uniqueness of zero matrices, and expressing matrices as sums of symmetric and skew-symmetric components.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_System/5.9E%3A_Exercises_for_Section_5.9This page discusses solutions to linear systems of equations in matrix form, providing detailed exercises along with their solution sets expressed as linear combinations of vectors. Each exercise illu...This page discusses solutions to linear systems of equations in matrix form, providing detailed exercises along with their solution sets expressed as linear combinations of vectors. Each exercise illustrates the relationship between homogeneous and non-homogeneous equations, presents the general solution format, and identifies the basis for the solution space.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEsThis page presents a curriculum on differential equations, addressing classification, integrals, slope fields, and both linear and nonlinear equations. It highlights solution existence and uniqueness,...This page presents a curriculum on differential equations, addressing classification, integrals, slope fields, and both linear and nonlinear equations. It highlights solution existence and uniqueness, along with first-order linear equations methods. Topics include separable and autonomous equations, numerical methods like Euler’s, and transforming nonlinear equations for easier manipulation. Each section is supported by exercises to enhance understanding.
