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**Math 3120 Differential Equations with Boundary Value Problems**

Chapter 1 Introduction to Differential Equations

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**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.

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**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

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**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,

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

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**Classifications of Differential Equation**

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

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**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:

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**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:

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Notation Leibniz Prime Dot Subscript

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**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.

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**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

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**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

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**Linear & Nonlinear Differential Equations**

Example: Determine whether the equations below are linear or nonlinear.

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**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.

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**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 (-∞, ∞).

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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.

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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.

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**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.

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**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)

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**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 are sufficient but not necessary.

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