Introduction 1. Classification of Differential Equations. Definition

Lecture 1: Introduction

1. Classiﬁcation of Diﬀerential Equations.

Definition: A Differential Equation (DE) is an equation that contains derivatives. Example: y′′ = x2, y′′ = y. Definition: An Ordinary DE is one with one independent variable only. Definition: A Partial DE is one with more than one independent variables, containing partial derivatives - this is the subject of Math 312. Definition: The Order of a DE is the order of the highest derivative present. Definition: A Linear Operator is an operator L[y] acting on a function that satisfies that (1) L[y1 + y2]= L[y1]+ L[y2] for any functions y1,y2 in the domain of L. (2) L[αy]= αL[y] for any α ∈ ℜ and any function y

Example 1: L[y]= y′′ + 5y′ − 2y is linear (check it) Example 2: L[y]= x3y′′ + (sin x)y is linear (check it) Example 3: L[y]= y2 is not linear (check it) b Example 4: L[y]= Ra y(x) dx is a linear integral operator (check it)

Deﬁnition: A Linear ODE is an equation of the form

L[y]= f(x)

where L[y] is a linear ordinary diﬀerential operator.

Example 1: x3y′′ + (sin x)y = x3 is a linear ODE Example 2: y3y′′ = 5x − 3 is not linear

In this class we will consider general ﬁrst and second order diﬀerential equations of the form

′ y = f(x,y)

′′ ′ y = f(x,y,y ) where, x is the independent variable. These may or may not be linear, depending on what f is.

2. Solutions to a DE. A solution to a differential equation, such as y′′ = f(t,y,y′), is a function y(t) that satisfies the equation. A general solution to a first order equation must have one constant of integration. A general solution to a second order equation must have two constants of integration. Example 1: Show that y = Cex solves y′ = y. Example 2: Show that y = ex + C does not solve y′ = y for any C. −x2 x t2 ′ Example 3: Show that y = e Ra e dt solves y = 1 − 2xy for any a

1 3. Goals for this semester. • DEs as mathematical models. Understand what the terms in a DE say about the physical process it is meant to model, for simple cases. Check that DEs are dimensionally correct. • Exact solutions. We’ll ﬁnd exact solutions for some ﬁrst order linear and nonlinear equations, and for second order linear equations, and second order linear systems of equations. • Qualitative behaviour of nonlinear systems. Including equilibria, long term behaviour and stability. • Numerical methods. Use numerical tools to visualize and solve ODEs, in MATLAB. • Some Theory. Existence and uniqueness of solutions. Linear operators. Linearity and the superposition principle. Use numerical tools to visualize and solve ODEs, in MATLAB.