Nonhomogeneous Equations and the Method of Undetermined Coefficients

date2026-07-16document_iddoc_03a23e75eacf5a680477ecfe9c14adc8description非同次線型微分方程式を、線型作用素 L の特殊解 + 核として整理し、未定係数法の条件と共鳴補正を説明する。prerequisites二階線型定数係数微分方程式の拡張理論 / 線型作用素方程式の基本type講義content_typelecturestatusactive

mathdifferential-equationsnonhomogeneouslecture

1Introduction

This lecture decomposes the general solution of a nonhomogeneous linear equation into a homogeneous solution and a particular solution and states the conditions under which the method of undetermined coefficients applies.

2Standard Form

Consider

L [y] = a y^{'}' + b y^{'} + c y = f (x),

with constant coefficients $a, b, c$ and $a \neq 0$ .

3Affine Structure of the Solution Set

Suppose that $L [y_{p}] = f$ and $L [y_{h}] = 0$ . Linearity gives

L [y_{h} + y_{p}] = f .

More precisely, if $h \in \ker L$ , then $L [y_{p} + h] = f$ . Conversely, if $L [y] = f$ , then $L [y - y_{p}] = 0$ , so $y - y_{p} \in \ker L$ . The complete solution set is therefore the affine space

y_{p} + \ker L .

The homogeneous solution space is $\ker L$ . This is the general structure of a linear operator equation.

data/lecture/math/linear-operator/linear-operator-equation-basics.lecture.n.md

4Conditions for the Method of Undetermined Coefficients

The {method of undetermined coefficients} is especially effective when the differential operator has constant coefficients and $f (x)$ is a polynomial, an exponential, a sine or cosine, or a finite sum or product of such functions. These function families lie in finite-dimensional spaces preserved by differentiation. A trial function must therefore span a differentiation-invariant function space rather than merely imitate the displayed forcing term.

Form of the forcing term	Basic form of a trial particular solution	Reason
$p_{n} (x)$	A polynomial of degree $n$	Differentiation only lowers the degree
$e^{α x}$	$A e^{α x}$	Differentiation produces a scalar multiple
$\cos β x, \sin β x$	$A \cos β x + B \sin β x$	Differentiation preserves their two-dimensional span
$e^{α x} p_{n} (x)$	$e^{α x} q_{n} (x)$	The exponential factor and polynomial space are preserved

5Example 1: Polynomial Forcing

Consider

y^{'}' - y = x .

The homogeneous solution is $y_{h} = C_{1} e^{x} + C_{2} e^{- x}$ . Because the right-hand side is a first-degree polynomial, set $y_{p} = A x + B$ . Then

y_{p^{'}}' - y_{p} = - A x - B .

Comparison with $x$ gives $A = - 1$ and $B = 0$ . Hence

y = C_{1} e^{x} + C_{2} e^{- x} - x .

The essential point is that the trial space spanned by $1, x$ is closed under differentiation.

6Resonance and the Trial-Function Correction

For

y^{'}' + y = \cos x,

the functions $\cos x$ and $\sin x$ belong to the homogeneous solution space. A trial function $A \cos x + B \sin x$ therefore lies in $\ker L$ and cannot produce the forcing term. Multiplication by $x$ removes this overlap, so use

y_{p} = x (A \cos x + B \sin x) .

Substitution gives one particular solution

y_{p} = \frac{1}{2} x \sin x .

Multiplication by $x$ is thus a linear-algebraic correction that moves the trial function outside the homogeneous solution space. In general, the required power of $x$ is determined by the multiplicity of the relevant characteristic root.

7Situations in Which the Method Does Not Apply Naturally

For

y^{'}' + y = \tan x,

the derivatives of $\tan x$ do not remain in a finite-dimensional trial space of the required type. The method of undetermined coefficients is therefore not natural, and variation of parameters is an appropriate alternative.

The constant-coefficient hypothesis is equally important. For an equation such as $y^{'}' + x y = f (x)$ , the variable coefficient destroys the invariant finite-dimensional structure even when the forcing term is simple.

8Comparison with Variation of Parameters

Method	Advantage	Restriction
Method of undetermined coefficients	Usually short computation	Strong dependence on constant coefficients and the form of the forcing term
Variation of parameters	Comparatively insensitive to the form of the forcing term	The resulting integrals may be difficult or non-elementary

Computational efficiency justifies trying undetermined coefficients first when its hypotheses hold. Once they fail, variation of parameters or a fundamental-matrix construction should be selected.

9Scope and Limitations

The method of undetermined coefficients is efficient precisely when a constant-coefficient linear differential operator preserves a suitable finite-dimensional function space. It fails when the forcing term has an incompatible form, coefficients vary with the independent variable, or overlap with the homogeneous solution space is left uncorrected. Failure of this method does not imply that the equation has no solution; variation of parameters often constructs a particular solution as an integral.

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13Related Lectures

data/lecture/math/linear-operator/linear-operator-equation-basics.lecture.n.md data/lecture/math/linear-operator/polynomial-operators-and-eigenvalue-problems.lecture.n.md data/lecture/math/linear-operator/heaviside-operator-method-and-transforms.lecture.n.md