## Fall 2021 lectures with steve butler

Introduction to differential equations -- Springing into action

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Verifying solutions; Initial value problems; Mathematical modeling

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Setting up a differential equation; Terminology; Examples of things that can go awry

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Simple physics examples; Existence and uniqueness

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Slope fields; Separable differential equations

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Practice with separable differential equations

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Method of integrating factor

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Substitution method

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Exact ODEs; Population models

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Population models; Autonomous differential equations

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Numerical approximation (Euler)

[Optional topic; not covered on exam/homework]

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nth-order differential equations; Wronskian and linear independence

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Solving constant coefficient homogeneous linear differential equations (1/2)

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Solving constant coefficient homogeneous linear differential equations (2/2)

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Nonhomogeneous strategy; Spring-mass systems

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Remaining material on spring-mass; short review of material

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Exam 1 practice / solutions

First practice exam (problems; solutions; walk-through)

Second practice exam (problems; solutions; walk-through)

Third practice exam (problems; solutions; walk-through)

In-class practice exam (problems; solutions; walk-through)

Fall 2021 exam (problems; solutions; walk-through)

Method of undetermined coefficients

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Method of variation of parameters

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Practice with variation of parameters; introduction to systems of differential equations

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Substitution and elimination to solve systems of differential equations

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Basics of matrices and linear algebra

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Basics of differential equations for systems

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Practice with systems of differential equations

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Eigenvalues and eigenvectors

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Complex eigenvalues; direction fields

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Handling missing eigenvectors (how to "bump" for systems)

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Fundamental matrix and matrix exponentials

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Comments on nonhomogeneous ODE systems [*optional material for Fall 2021*]

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Introduction to the Laplace transform

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Laplace transforms and derivatives

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Laplace transforms and partial fractions

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More rules for Laplace transforms

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The 𝛿-function and dealing with shifts

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Additional practice with Laplace transforms

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Review of material for midterm 2

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Exam 2 practice / solutions

[Butler] First practice exam (problems; solutions; walk-through)

[Butler] Second practice exam (problems; solutions; walk-through)

[Butler] Third practice exam (problems; solutions; walk-through)

[Basnet] Fourth practice exam (problems; solutions; walk-through)

[Butler] In-class practice exam (problems; solutions; walk-through)

Fall 2021 exam (problems; solutions; walk-through)

Introduction to using power series to solve differential equations

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Radius of convergence of solutions

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Ordinary/singular points

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Review

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Final exam practice / solutions

[Basnet] First practice exam (problems; solutions; walk-through)

[Basnet] Second practice exam (problems; solutions; walk-through)

[Butler] Third practice exam (problems; solutions; walk-through)

[Butler] Fourth practice exam (problems; solutions; walk-through)

Fall 2021 exam (solutions; walk-through)