Calculus and Differential Equations I

A simple diﬀerential equation

Is there a function which is equal to its derivative? 1 Yes Calculus and Differential Equations I 2 No MATH 250 A Is such a function unique? 1 Yes 2 No Introduction to differential equations dy The equation = y is an example of an ordinary differential dx equation.

The independent variable is x and the dependent variable is y.

The above equation is a ﬁrst order, autonomous and linear diﬀerential equation.

Introduction to differential equations Calculus and Differential Equations I Introduction to differential equations Calculus and Differential Equations I

Solutions of a diﬀerential equation The nonlinear pendulum

An explicit solution of an ordinary differential equation The equation of motion for the dy nonlinear pendulum is given by = f (x, y) dx d2θ dθ ml = −mg sin(θ)−cl , is a function y(x) such that when substituted into the dt2 dt differential equation, both sides are found to be identical. →j l where In general, a given differential equation will have a family of ı→ θ and t are variables. solutions, involving one or more parameters. → θ m, l, g and c are Applying initial or boundary conditions often leads to the θ m selection of one of these solutions. parameters. r→ We will now turn to two important questions: what are g→ Most of these quantities differential equations used for,andhow do we study them? are defined on the figure, Sketch of a point-mass pendulum except c, which measures To address the first questions, we now look at examples of friction. differential equations or systems thereof.

Introduction to differential equations Calculus and Differential Equations I Introduction to differential equations Calculus and Differential Equations I The RLC circuit The classic SIR model

The classic SIR model reads The series RLC circuit consists of a dS I resistor of resistance R,aninductor = −α S , of inductance L,acapacitor of dt N dI I capacitance C, and a power source = α S − β I , of voltage V (t). dt N dR = β I , The charge q across the capacitor dt satisﬁes the diﬀerential equation where S, I ,andR represent the d2q dq 1 numbers of susceptible, infectious,and L + R + q = V (t), dt2 dt C recovered (or removed) individuals, in a Image by Omegatron released population of size N. The parameter α under a Creative Commons and the current in the circuit is given Attribution ShareAlike license dq measures the average number of versions 3.0, 2.5, 2.0, and 1.0 by I (t)= . positive contacts per susceptible per dt Penned goats in a village within a region investigated for a Rift Valley fever outbreak unit of time, and β measures the rate at in Saudi Arabia. Picture # 8362, Public which individuals recover. Health Image Library

Introduction to differential equations Calculus and Differential Equations I Introduction to differential equations Calculus and Differential Equations I

Viral infections How do we study diﬀerential equations?

The the dynamics of a viral infection, such as hepatitis B or C, may be Sometimes, we can solve a differential equation. In this class described by the following model (M.A. (MATH 250 A & B), we will learn how to solve first and Nowak et al., Proc. Natl. Acad. Sci. second order linear equations and systems of first order linear USA 93, 4398-4402 (1996)). equations,aswellassome first order nonlinear equations.

dX If initial conditions are known, one can solve a differential = λ − δ X − bV X dt equation (or a system of differential equations) numerically. dY = bV X − aY We will learn a simple numerical method to solve a differential dt equation and also use more advanced algorithms in MATLAB. dV = kY − κ V dt Before trying to solve a differential equation, or launching into The variable X represents the number a numerical exploration of its properties, one needs to know Transmission electron micrograph showing of uninfected cells, Y is the number of whether solutions exist and if so, whether they are unique.We hepatitis virions of an unknown strain. will see theorems that guarantee existence and uniqueness of Picture # 8153, Public Health Image infected cells,andV is the viral load (or Library number of free virions in the body). solutions to differential equations.

Introduction to differential equations Calculus and Differential Equations I Introduction to differential equations Calculus and Differential Equations I How do we study differential equations? (continued) What we will do next

We will start with the simplest type of differential equations, In many situations, especially when one deals with nonlinear dy differential equations, one cannot find explicit solutions. = g(x). dx In this case, one can nevertheless understand the dynamics of Chapter 1 of Differential Equations book. Reading assignment: a differential equations by looking at special solutions and at Sections 1.1, 1.2, and 1.3. their stability. Solving such differential equations involves integration, so we will introduce various methods of integration discussed in the The qualitative theory of dynamical systems (discussed in Calculus book. MATH 454) provides a way of understanding the behavior of a system of differential equations, as well as the bifurcations We will then consider autonomous differential equations of the that occur when one or more parameters are changed. We will dy form = g(y). briefly address some of these issues. dx Partial differential equations (discussed in MATH 322, MATH Chapter 2 of the Differential Equations book. 422, and MATH 456) are differential equations describing the Ideas of stability and bifurcations. dynamics of systems with two or more independent variables. One more technique of integration: partial fractions.

Introduction to differential equations Calculus and Differential Equations I Introduction to differential equations Calculus and Differential Equations I

What we will do next (continued)

We will then turn to general first order differential equations, dy = g(x, y) dx Chapter 3 of the Differential Equations book. Graphical analysis, symmetries, scalings,andnumerical solutions. Existence and uniqueness of solutions.

Finally, we will look at various methods of solution for first order differential equations Chapters 4 and 5 of the Differential Equations book. Separation of variables and equations with homogeneous coefficients. First order linear differential equations and Bernouilli’s equation.

Introduction to diﬀerential equations Calculus and Diﬀerential Equations I