ch 4 3 nonhomogeneous equations method of undetermined coefficients l.
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Ch 4.3: Nonhomogeneous Equations: Method of Undetermined Coefficients. The method of undetermined coefficients can be used to find a particular solution Y of an n th order linear, constant coefficient, nonhomogeneous ODE provided g is of an appropriate form.

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ch 4 3 nonhomogeneous equations method of undetermined coefficients
Ch 4.3: Nonhomogeneous Equations: Method of Undetermined Coefficients
  • The method of undetermined coefficients can be used to find a particular solution Y of an nth order linear, constant coefficient, nonhomogeneous ODE

provided g is of an appropriate form.

  • As with 2nd order equations, the method of undetermined coefficients is typically used when g is a sum or product of polynomial, exponential, and sine or cosine functions.
  • Section 4.4 discusses the more general variation of parameters method.
example 1
Example 1
  • Consider the differential equation
  • For the homogeneous case,
  • Thus the general solution of homogeneous equation is
  • For nonhomogeneous case, keep in mind the form of homogeneous solution. Thus begin with
  • As in Chapter 3, it can be shown that
example 2
Example 2
  • Consider the equation
  • For the homogeneous case,
  • Thus the general solution of homogeneous equation is
  • For the nonhomogeneous case, begin with
  • As in Chapter 3, it can be shown that
example 3
Example 3
  • Consider the equation
  • As in Example 2, the general solution of homogeneous equation is
  • For the nonhomogeneous case, begin with
  • As in Chapter 3, it can be shown that
example 4
Example 4
  • Consider the equation
  • For the homogeneous case,
  • Thus the general solution of homogeneous equation is
  • For nonhomogeneous case, keep in mind form of homogeneous solution. Thus we have two subcases:
  • As in Chapter 3, can be shown that