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    Week1 Basic Concept

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    Shinta Arvinda P. Wulandari
    The document discusses various types of first-order differential equations, including linear equations, Bernoulli equations, homogeneous equations, separable equations, and exact equations. It defines each type of equation, provides examples of the standard form for each, and describes methods for solving them.

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    Week1 Basic Concept

    Uploaded by

    Shinta Arvinda P. Wulandari
    164 views21 pages
    The document discusses various types of first-order differential equations, including linear equations, Bernoulli equations, homogeneous equations, separable equations, and exact equations. It defines each type of equation, provides examples of the standard form for each, and describes methods for solving them.

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    The document discusses various types of first-order differential equations, including linear equations, Bernoulli equations, homogeneous equations, separable equations, and exact equations. It defines each type of equation, provides examples of the standard form for each, and describes methods for solving them.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    164 views21 pages

    Week1 Basic Concept

    Uploaded by

    Shinta Arvinda P. Wulandari
    The document discusses various types of first-order differential equations, including linear equations, Bernoulli equations, homogeneous equations, separable equations, and exact equations. It defines each type of equation, provides examples of the standard form for each, and describes methods for solving them.

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    © All Rights Reserved
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    TK-124 Matematika Teknik Kimia I

    Basic Concepts and Classifying


    Differential Equations

    First Order Differential Equations

    Dicky Dermawan
    dickydermawan@gmail.com
    Modeling with Differential Equations

    Perhaps the most important of all the applications of


    calculus
    is differential equations.

    When physical scientists or social scientists use


    calculus, more often than not it is to analyze a
    differential equation that has arisen in the process of
    modeling some phenomenon that they are studying.

    Although it is often impossible to find an explicit


    formula for the solution of a differential equation, we
    will see that graphical and numerical approaches
    Differential Equations
    A differential equation is an equation involving an
    unknown function and its derivatives.
    A differential
    equation is an
    ordinary differential
    equation if the
    unknown
    function depends on
    only one
    independent
    variable. If the
    unknown function
    depends on two or
    more independent
    Order & Solution
    The order of a differential equation is the order of the
    highest derivative appearing in the equation.
    A solution of a differential equation in the unknown
    function y and the independent variable x on the
    interval is a function y(x) that satisfies the differential
    equation identically for all x in .

    Some differential equations have infinitely many solutions,


    whereas other differential equations have no solutions. It is
    also possible that a differential equation has exactly one
    solution.
    The general solution of a differential equation is the set of
    all solutions.
    Initial-Value vs Boundary-Value
    Problems
    A differential equation along with subsidiary conditions
    on the unknown function and its derivatives, all given
    at the same value of the independent variable,
    constitutes an initial-value problem. The subsidiary
    conditions are initial conditions.

    If the subsidiary conditions are given at more than one


    value of the independent variable, the problem is a
    boundary-value problem
    and the conditions are boundary conditions.
    Standard and Differential
    Forms
    Standard form for a first-order differential equation in
    the unknown function y(x) is

    Differential form
    First-Order Differential Equations
    Linear Equations
    Bernoulli Equations
    Homogeneous Equations
    Separable Equations
    Exact Equations
    Linear Equations
    A differential equation in standard form

    is linear if f(x,y)= p(x)y+q(x). First-order linear


    differential equations can always be expressed as

    An integrating factor for Equation 1.8 is

    which depends only on x and is independent of y.


    When both sides of 1.8 are multiplied by I(x), the
    resulting equation
    Bernoulli Equations
    A Bernoulli differential equation is an equation of the
    form

    where n denotes a real number.


    When n = 1 or n = 0, a Bernoulli equation reduces to a
    linear equation.
    Homogeneous Equations
    A differential equation in standard form

    is homogeneous if

    for every real number t.


    In the general framework of differential equations, the
    word homogeneous has an entirely different
    meaning. Only in the context of first-order differential
    equations does homogeneous have the meaning
    defined above.
    Soal
    Separable Equations
    Consider a differential equation in differential form:

    If M(x,y) = A(x) (a function only of x) and N(x,y) = B(y)


    (a function only of y), the differential equation is
    separable, or has its variables separated.
    The general solution to the first-order separable
    differential equation:

    is

    where c represents an arbitrary constant.


    Exact Equations
    A differential equation in differential form

    is exact if

    Method of Solution: first solve the equations

    and

    for g(x, y). The solution is then given implicitly by


    Determine the ODEs classification
    Write the given differential equations in
    standard form

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