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    Undetermined Coefficients and Cauchy-Euler

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    Lief Deltora
    The document discusses methods for solving higher-order linear differential equations with constant coefficients, including nonhomogeneous and Cauchy-Euler equations. It describes solving nonhomogeneous equations by finding the complementary solution to the associated homogeneous equation and any particular solution. The method of undetermined coefficients is discussed for cases where terms in the particular solution are or aren't solutions to the homogeneous equation. Cauchy-Euler equations are defined and solved by substituting solutions of the form xm, finding the auxiliary equation, and determining the form of the general solution based on the roots. Examples are provided to illustrate the methods.

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    © All Rights Reserved
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    Undetermined Coefficients and Cauchy-Euler

    Uploaded by

    Lief Deltora
    219 views16 pages
    The document discusses methods for solving higher-order linear differential equations with constant coefficients, including nonhomogeneous and Cauchy-Euler equations. It describes solving nonhomogeneous equations by finding the complementary solution to the associated homogeneous equation and any particular solution. The method of undetermined coefficients is discussed for cases where terms in the particular solution are or aren't solutions to the homogeneous equation. Cauchy-Euler equations are defined and solved by substituting solutions of the form xm, finding the auxiliary equation, and determining the form of the general solution based on the roots. Examples are provided to illustrate the methods.

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    3. Undetermined Coefficients and Cauchy-Euler.pptx

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    The document discusses methods for solving higher-order linear differential equations with constant coefficients, including nonhomogeneous and Cauchy-Euler equations. It describes solving nonhomogeneous equations by finding the complementary solution to the associated homogeneous equation and any particular solution. The method of undetermined coefficients is discussed for cases where terms in the particular solution are or aren't solutions to the homogeneous equation. Cauchy-Euler equations are defined and solved by substituting solutions of the form xm, finding the auxiliary equation, and determining the form of the general solution based on the roots. Examples are provided to illustrate the methods.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    219 views16 pages

    Undetermined Coefficients and Cauchy-Euler

    Uploaded by

    Lief Deltora
    The document discusses methods for solving higher-order linear differential equations with constant coefficients, including nonhomogeneous and Cauchy-Euler equations. It describes solving nonhomogeneous equations by finding the complementary solution to the associated homogeneous equation and any particular solution. The method of undetermined coefficients is discussed for cases where terms in the particular solution are or aren't solutions to the homogeneous equation. Cauchy-Euler equations are defined and solved by substituting solutions of the form xm, finding the auxiliary equation, and determining the form of the general solution based on the roots. Examples are provided to illustrate the methods.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 16

    Higher-Order Differential

    Equations
    Linear Nonhomogeneous Differential Equations with Constant
    Coefficients
    Nonhomogeneous Linear DE
    To solve a nonhomogeneous linear differential equation,

    a
    ny(
    n )
    a
    n
    1y(
    n 1)
     
    ...
    a y
    1 '
    a0yg
    (x)
    we must do two things:
    1.) Find the complementary function yc
    2.) Find any particular solution yp of the nonhomogeneous equation.

    The general solution of the nonhomogeneous linear DE on an interval I


    is y = yc + yp
    Nonhomogeneous Linear DE
    The complementary function yc is the general solution of the
    associated homogeneous DE of the original nonhomogeneous
    differential equation, that is

    an y ( n )  an1 y ( n1)  ...  a1 y ' a0 y  0


    Method of Undetermined Coefficients
    The underlying idea in this method is a conjecture, an educated guess
    really, about the form of yp motivated by the kinds of functions that
    make up the input function g(x). The general method is limited to non-

    homogeneous linear DEs such as n a y(n)
    an1y (
    n 1)
     
    ...a1y'a0 y g (x)
    where:
    • the coefficients a0, a1, ..., an are constants, and
    • g(x) is either a constant, a polynomial function, an exponential
    function eαx, a sine function sin βx, a cosine function cos βx, or finite
    sums and products of these functions
    Method of Undetermined Coefficients
    Method of Undetermined Coefficients
    Case 1
    No function in the assumed particular solution is a solution of
    the associated homogeneous differential equation.
    Example:
    y”-8y'+25y = 5x3e-x - 7e-x

    y”-8y'+25y = (5x3 - 7)e-x


    yc = e4x (c1cos 3x + c2sin 3x)
    yp = (Ax3 + Bx2 + Cx + E)e-x
    No duplication between terms in yp and the terms in yc
    Method of Undetermined Coefficients
    Case 2
    A function in the assumed particular solution is also a solution of
    the associated homogeneous differential equation.
    Example:
    y”-5y'+4y = 8ex

    yc = c1ex + c2e4x
    yp = Aex
    Substituting to the differential equation yields 0 = 8ex. Use yp = Axex
    Use yp = Axn+1eαx for yc containing cxneαx to eliminate all duplications.
    Example 1 (Undetermined Coeffieients Case 1)
    Solve y” - 2y' - 3y = 4x - 5 + 6xe2x

    Ans: y = c1e-x + c2e3x - 4x/3 + 23/9 - (2x + 4/3)e2x


    Example 2 (Undetermined Coefficients Case 2)
    Solve the IVP y” + y = 4x + 10 sin x , y(π) = 0, y'(π) = 2

    Ans: y = 9π cos x + 7 sin x + 4x - 5x cos x


    Cauchy-Euler Equations
    Any linear differential equation of the form
    an x n y ( n )  an1 x n1 y ( n1)  ...  a1 xy ' a0 y  g ( x)
    where the coefficients an, an-1, ..., a0 are constants, is known as a
    Cauchy-Euler equation in honor of and
    . The observable characteristic of this type of equation
    is that the degree k = n, n-1, ..., 1, 0 of the monomial coefficients xk
    matches the order of differentiation dky/dxk.
    Cauchy-Euler Equations
    Note:
    The coefficient of dny/dxn is zero at x=0. Hence, we confine our
    attention to finding the general solution on the interval (0,∞). Solutions
    on the interval (-∞,0) can be obtained by substituting a variable t = -x
    into the differential equation.
    Method of Solution
    We try a solution of the form y = xm where m is to be determined.
    Analogous to what happened when we substitute emx into a linear
    equation with constant coefficients, after substituting xm, each term of
    a Cauchy-Euler equation becomes a polynomial in m times xm. For
    example, by substituting y = xm to a second-order DE, it becomes
    y  xm
    y '  mxm1
    y ' '  m(m  1) x m2
    a2 x 2 y ' ' a1xy ' a0 y  a2 x 2 m(m  1) x m2  a1 xmxm1  a0 x m
     a2 m(m  1) x m  a1mxm  a0 x m
    Method of Solution
    Thus y = xm is a solution of the differential equation whenever m is a
    solution of the auxillary equation a2m(m-1) + a1m + a0 = 0.
    Case 1. Distinct Real Roots (m1 ≠ m2)
    y = c1xm1 + c2xm2

    Case 2. Repeated Real Roots (m1 = m2)


    y = c1xm1 + c2xm1 ln x

    Case 3. Conjugate Complex Roots (m = α ± βi)


    y = xα [c1 cos(β ln x) + c2 sin((β ln x)]
    Example 3 (Cauchy-Euler Case 1)
    Solve: x2y'' - 2xy' - 4y = 0

    Ans: y = c1x-1 + c2x4


    Example 4 (Cauchy-Euler Case 2)
    Solve: 4x2y'' + 8xy' + y = 0

    Ans: y = c1x-1/2 + c2x-1/2 ln x


    Example 5 (Cauchy-Euler Case 3)
    Solve the IVP: 4x2y'' + 17y = 0, y(1) = -1, y'(1) = -1/2

    Ans: y = -x1/2 cos (2 ln x)

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