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- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_222_Calculus_2/03%3A_Introduction_to_Differential_Equations/3.05%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/04%3A_Generating_Functions/4.03%3A_Generating_Functions_and_Recurrence_RelationsAlgebraic manipulations with generating functions can sometimes reveal the solutions to a recurrence relation.
- 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/Mission_College/MAT_3B_Calculus_II_(Kravets)/06%3A_Introduction_to_Differential_Equations/6.05%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/11%3A_Introduction_to_Differential_Equations/11.06%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Tran)/05%3A_Introduction_to_Differential_Equations/5.06%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/08%3A_Introduction_to_Differential_Equations/8.05%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_RBy generating all linear combinations of a set of vectors one can obtain various subsets of \(\mathbb{R}^{n}\) which we call subspaces. For example what set of vectors in \(\mathbb{R}^{3}\) generate t...By generating all linear combinations of a set of vectors one can obtain various subsets of \(\mathbb{R}^{n}\) which we call subspaces. For example what set of vectors in \(\mathbb{R}^{3}\) generate the \(XY\)-plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.03%3A_Elementary_MatricesThis page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expre...This page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expresses a matrix as a product of elementary matrices. Properties of invertible matrices are discussed, including the conditions that an \(n \times n\) matrix must meet to be invertible, emphasizing the significance of row operations.
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Everett)/05%3A_Introduction_to_Differential_Equations/5.06%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/08%3A_Introduction_to_Differential_Equations/8.05%3A_First-order_Linear_EquationsAny first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may ...Any first-order linear differential equation can be written in the form y′+p(x)y=q(x). We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
