Presentation on theme: "Math 3120 Differential Equations with Boundary Value Problems"— Presentation transcript:
1 Math 3120 Differential Equations with Boundary Value Problems Chapter 1Introduction to Differential Equations
2 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.
3 Basic Mathematical Models Formulate a mathematical model describing motion of an object falling in the atmosphere near sea level.Variables: time t, velocity vNewton’s 2nd Law: F = ma = net forceForce of gravity: F = mg downward forceForce of air resistance: F = v upward forceThen
4 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 releaseThen,
6 Classifications of Differential Equation By TypesOrdinary Differential Equation (ODE)Partial Differential Equation (PDE)OrderSystemsLinearityLinearNon-Linear
7 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:
8 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:
10 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 formwhere 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.
11 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 asThe normal form of Eq. (6) is
12 Linear & Nonlinear Differential Equations An ordinary differential equationis linear if F is linear in the variablesThus the general linear ODE has the formThe characteristic of linear ODE is given as
13 Linear & Nonlinear Differential Equations Example: Determine whether the equations below are linear or nonlinear.
14 Solutions to Differential Equations A solution of an ordinary differential equationon an interval I is a function (t) such thatexists 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 solutionsNOTE: Solutions of ODE are always defined on an interval.
15 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 (-∞, ∞).
16 Types of SolutionsTrivial 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 ODEImplicit solution is a solution that is not in explicit form.
17 Families of SolutionsA solution of a first- order differential equationusually 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.
18 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 typeAn nth order IVP is of the formwhere are arbitrary constants.Note: The number of IC’s depend on the order of the DE.
19 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)
20 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 IVPIt turns out that conditions stated in Theorem are sufficient but not necessary.