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    Intro ODE

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    Fikrian
    The document provides an introduction to differential equations, covering definitions, classifications, and initial-value problems. It distinguishes between ordinary and partial differential equations, as well as explicit and implicit solutions. Additionally, it discusses the role of differential equations as mathematical models in various applications.

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    Intro ODE

    Uploaded by

    Fikrian
    3 views19 pages
    The document provides an introduction to differential equations, covering definitions, classifications, and initial-value problems. It distinguishes between ordinary and partial differential equations, as well as explicit and implicit solutions. Additionally, it discusses the role of differential equations as mathematical models in various applications.

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    © © All Rights Reserved

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    The document provides an introduction to differential equations, covering definitions, classifications, and initial-value problems. It distinguishes between ordinary and partial differential equations, as well as explicit and implicit solutions. Additionally, it discusses the role of differential equations as mathematical models in various applications.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    3 views19 pages

    Intro ODE

    Uploaded by

    Fikrian
    The document provides an introduction to differential equations, covering definitions, classifications, and initial-value problems. It distinguishes between ordinary and partial differential equations, as well as explicit and implicit solutions. Additionally, it discusses the role of differential equations as mathematical models in various applications.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    You are on page 1of 19

    INTRODUCTION TO

    DIFFERENTIAL EQUATIONS

    Dr . Imam Jauhari Maknun, S.T. M.T., MSc.


    OUTLINE

    1. Definitions and Terminology

    2. Initial-Value Problems

    3. Differential Equations as Mathematical Models


    Definitions and Terminology
    Introduction
    The words differential and equation certainly suggest
    solving some kind of equation that contains
    derivatives.

    But before you start solving anything, you must learn


    some of the basic definitions and terminology of the
    subject
    Definition
    The derivative dy/dx of a function y = ϕ (x) is itself another
    function ϕ '(x) found by an appropriate rule.
    For example, the function is differentiable on
    the interval (- ∞, ∞), and its derivative is
    If we replace in the last equation by the symbol
    y, we obtain
    Classification by Type
    If a differential equation contains only ordinary derivatives of
    one or more functions with respect to a single independent
    variable it is said to be an ordinary differential equation
    (ODE).
    Classification by Type
    An equation involving only partial derivatives of
    one or more functions of two or more independent
    variables is called a partial differential equation
    (PDE).
    Notation
    Throughout this text, ordinary derivatives will be written
    using either the Leibniz notation dy/dx, d2y/dx2, d3y/dx3, • •
    • , or the prime notation y', y", y"', ....
    Classification by Order
    The order of a differential equation (ODE or PDE) is the order
    of the highest derivative in the equation.
    The differential equations

    are examples of a second-order ordinary differential equation


    and a fourth-order partial differential equation, respectively.
    Classification by Linearity
     The equations

    are, in turn, examples of linear first-, second-, and third-order


    ordinary differential equations.
     The equations

    are examples of nonlinear first-, second-, and fourth-order ordinary


    differential equations, respectively
    Solution
    Explicit and Implicit Solutions
     You should be familiar with the terms explicit and implicit
    functions from your study of calculus.
     A solution in which the dependent variable is expressed
    solely in terms of the independent variable and constants
    is said to be an explicit solution.
    Systems of Differential Equations
     Up to this point we have been discussing single differential
    equations containing one unknown function.

     But often in theory, as well as in many applications, we must


    deal with systems of differential equations
    Initial-Value Problems
    Initial-Value Problem
    We are often interested in problems in which we seek a
    solution y(x) of a differential equation so that y(x) satisfies
    prescribed side conditions-that is, conditions that are
    imposed on the unknown y(x) or on its derivatives
    Differential Equations as
    Mathematical Models
    Mathematical Models
    Steps in the modeling process
    Mathematical Models

    Mixing tank Water draining from a tank Position of rock measured


    from ground level
    Mathematical Models

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