# Lecture Notes From Introduction to ODEs and PDEs

These notes were formed and crafted from teaching Math 251: Introduction to ODEs and PDEs at Penn State University from 2010-2014. The course followed the textbook of Boyce and DePrima 9th Edition. Feel free to use these lecture notes to help you learn the material, but they are for personal use only (no sale or distribution). If any typos or errors are observed please let me know.

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1.1 Direction Fields
1.2 Solutions to Some Diff. Eqns
1.3 Classifications of Diff. Eqns
2.1 Linear Equations with Variable Coefficients
2.2 Separable Equations
2.3 Modeling with First Order Eqns
2.4 Differences Between Linear and Nonlinear Eqns.
2.5 Autonomous Eqns. and Population Dynamics
2.6 Exact Equations
3.1 Homogeneous Eqns. with Constant Coefficients
3.2 Fundamental Solutions of Linear Homogeneous Eqns.
3.3 Complex Roots and Characteristic Eqn.
3.4 Repeated Roots and Reduction of Order
3.5 Non-homogeneous Eqns., Undetermined Coeff.
3.7 Electrical and Mechanical Vibrations
3.8 Forced Vibrations without Damping
4.1 General Theory for nth Order Eqns.
4.2 nth Order Homogeneous Eqns. with Constant Coefficients
6.1 Definition of Laplace Transform
6.2 Solutions of Initial Value Problems
6.3 Step Functions
6.4 Diff. Eqns. with Discontinuous Forcing Functions
6.5 Direc Delta and Laplace Transforms
7.1-7.2 Introduction to Systems of Diff. Eqns.
7.3 Systems of Linear Algebraic Equations
7.5 Homogeneous Linear Systems Constant Coefficients
7.6 Complex Eigenvalues
7.8 Repeated Eigenvalues
9.1 Phase Plane
9.2 Autonomous Systems, Linear Stability, 9.3 Almost Linear Systems
9.5 Predator Prey Equations
10.1 Two Point Boundary Value Problems
10.2 Fourier Series
10.3-10.4 Preview
10.3 Fourier Convergence Theorem
10.4 Even and Odd Functions
10.5 Separation of Variables
10.6 Other Heat Conduction Problems
10.7 Wave Equation and Vibrations of an Elastic String
10.8 Laplace's Equation

## Current Research

• Modeling, Analysis, and Simulation for Problems in Biology, Medicine, and Materials Science

• Computational and Applied Mathematics

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• Applied Analysis

• Collective Dynamics, Kinetic Theory

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Gil Ariel (Bar Ilan)

Igor Aronson (ANL)

Orkun Baluglu, MD (Cle Clinic)

Avraham Be'er (Ben-Gurion)

Leonid Berlyand (PSU)

Dmitry Golovaty (U. Akron)

Brian Haines (LANL)

Dmitry Karpeev (ANL)

Zach McCarthy (York U.)

Dan Munther (CSU)

Peter Palffy-Muhoray (Kent St)

Misha Potomkin (PSU)

Valerii Vinokur (ANL)

J. Patrick Wilber (U. Akron)

Chuan Xue (OSU)

Xiaoyu Zheng (Kent St)