Math 3120 Differential Equations with Boundary Value Problems Chapter 1 Introduction to Differential Equations

Basic Mathematical Models Many physical systems describing the real world are statements or relations involving rate of change. In mathematical terms, statements are equations and rates are derivatives. Definition: An equation containing derivatives is called a differential equation. Differential equation (DE) play a prominent role in physics, engineering, chemistry, biology and other disciplines. For example: Motion of fluids, Flow of current in electrical circuits, Dissipation of heat in solid objects, Seismic waves, Population dynamics etc. Definition: A differential equation that describes a physical process is often called a mathematical model.

Basic Mathematical Models Formulate a mathematical model describing motion of an object falling in the atmosphere near sea level. Variables: time t, velocity v Newton’s 2nd Law: F = ma = net force Force of gravity: F = mg downward force Force of air resistance: F =  v upward force Then

Basic Mathematical Models We can also write Newton’s 2nd Law: where s(t) is the distance the body falls in time t from its initial point of release Then,

Examples of DE (1) (2) (3) (4) (5)

Classifications of Differential Equation By Types Ordinary Differential Equation (ODE) Partial Differential Equation (PDE) Order Systems Linearity Linear Non-Linear

Ordinary Differential Equations When the unknown function depends on a single independent variable, only ordinary derivatives appear in the equation. In this case the equation is said to be an ordinary differential equations. For example: A DE can contain more than one dependent variable. For example:

Partial Differential Equations When the unknown function depends on several independent variables, partial derivatives appear in the equation. In this case the equation is said to be a partial differential equation. Examples:

Notation Leibniz Prime Dot Subscript

Systems of Differential Equations Another classification of differential equations depends on the number of unknown functions that are involved. If there is a single unknown function to be found, then one equation is sufficient. If there are two or more unknown functions, then a system of equations is required. For example, Lotka-Volterra (predator-prey) equations have the form where u(t) and v(t) are the respective populations of prey and predator species. The constants a, c, ,  depend on the particular species being studied.

Order of Differential Equations The order of a differential equation is the order of the highest derivative that appears in the equation. Examples: An nth order differential equation can be written as The normal form of Eq. (6) is

Linear & Nonlinear Differential Equations An ordinary differential equation is linear if F is linear in the variables Thus the general linear ODE has the form The characteristic of linear ODE is given as

Linear & Nonlinear Differential Equations Example: Determine whether the equations below are linear or nonlinear.

Solutions to Differential Equations A solution of an ordinary differential equation on an interval I is a function (t) such that exists and satisfies the equation: for every t in I. Unless stated we shall assume that function f of Eq. (7) is a real valued function and we are interested in obtaining real valued solutions NOTE: Solutions of ODE are always defined on an interval.

Solutions to Differential Equations Example: Show that is a solution of the ODE on the interval (-∞, ∞). Verify that is a solutions of the ODE on the interval (-∞, ∞).

Types of Solutions Trivial solution: is a solution of a differential equation that is identically zero on an interval I. Explicit solution: is a solution in which the dependent variable is expressed solely in terms of the independent variable and constants. For example, are two explicit solutions of the ODE Implicit solution is a solution that is not in explicit form.

Families of Solutions A solution of a first- order differential equation usually contains a single arbitrary constant or parameter c. One-parameter family of solution: is a solution containing an arbitrary constant represented by a set of solutions. Particular solution: is a solution of a differential equation that is free of arbitrary parameters.

Initial Value Problems (IVP) Initial Conditions (IC) are values of the solution and /or its derivatives at specific points on the given interval I. A differential equation along with an appropriate number of IC is called an initial value problem. Generally, a first order differential equation is of the type An nth order IVP is of the form where are arbitrary constants. Note: The number of IC’s depend on the order of the DE.

Solutions to Differential Equations Three important questions in the study of differential equations: Is there a solution? (Existence) If there is a solution, is it unique? (Uniqueness) If there is a solution, how do we find it? (Qualitative Solution, Analytical Solution, Numerical Approximation)

Theorem 1.2.1: Existence of a Unique Solution Suppose f and f/y are continuous on some open rectangle R defined by (t, y)  (,  ) x (,  ) containing the point (t0, y0). Then in some interval (t0 - h, t0 + h)  (,  ) there exists a unique solution y = (t) that satisfies the IVP It turns out that conditions stated in Theorem 1.2.1 are sufficient but not necessary.