Non-Homogeneous Linear Equations: Undetermined Coefficients & Variation of Parameters

1. Theorem on the Non-Homogeneous Solution Structure

Let $L[y] = g(x)$ be an $n$-th order linear differential equation, and let $L[y] = 0$ be its associated homogeneous equation. If $y_p(x)$ is any particular solution of $L[y] = g(x)$, and $\{y_1, y_2, \dots, y_n\}$ is a fundamental solution set of $L[y] = 0$, then the complete general solution is:

where $y_c(x)$ is the complementary function and $y_p(x)$ is the particular integral.

2. Extended Superposition for Multiple Forcing Functions

If $g(x) = g_1(x) + g_2(x) + \dots + g_m(x)$, and $y_{p,i}(x)$ is a particular solution to $L[y] = g_i(x)$ for each $i = 1, \dots, m$, then by linearity of $L$:

Hence, the total particular solution is the sum $y_p(x) = \sum_{i=1}^m y_{p,i}(x)$.

1. Scope and Trial Forms for Undetermined Coefficients

The method of undetermined coefficients applies to constant-coefficient linear ODEs where the forcing function $g(x)$ is a linear combination of polynomials, exponentials, sines, and cosines:

2. The Modification Rule

The Modification Rule: If any term in the trial form of $y_p(x)$ is a solution of the associated homogeneous equation (i.e., appears in $y_c(x)$), that trial form must be multiplied by $x^s$, where $s$ is the smallest positive integer such that no term in $x^s y_p(x)$ belongs to $y_c(x)$. In other words, $s$ is the multiplicity of the root in the characteristic equation.

3. The Annihilator Operator Method

Let $D = \frac{d}{dx}$ denote the differential operator. An operator $A(D)$ is an annihilator for $g(x)$ if $A(D)[g(x)] \equiv 0$.

• $D^n$ annihilates $1, x, x^2, \dots, x^{n-1}$.
• $(D - \alpha)^n$ annihilates $e^{\alpha x}, x e^{\alpha x}, \dots, x^{n-1} e^{\alpha x}$.
• $[(D - \alpha)^2 + \beta^2]^n$ annihilates $x^k e^{\alpha x}\cos\beta x$ and $x^k e^{\alpha x}\sin\beta x$ for $k = 0, 1, \dots, n-1$.

If $L(D)[y] = g(x)$ and $A(D)[g] = 0$, applying $A(D)$ to both sides gives the higher-order homogeneous equation $A(D)L(D)[y] = 0$. Solving this homogeneous equation immediately reveals both $y_c(x)$ and the exact required trial form for $y_p(x)$.

1. Motivation and Scope

The method of undetermined coefficients is restricted to polynomials, exponentials, and sinusoidal forcing. When $g(x)$ is any general continuous function (e.g., $\tan x, \sec x, \frac{1}{x}, \ln x$), Lagrange's Method of Variation of Parameters provides a universal, exact integral formula.

2. Rigorous Derivation for Second-Order Equations

Consider $y'' + P(x) y' + Q(x) y = g(x)$, with known complementary solution $y_c(x) = c_1 y_1(x) + c_2 y_2(x)$. We replace the constants $c_1, c_2$ with functions $u_1(x), u_2(x)$:

Differentiating with respect to $x$:

To avoid second derivatives of $u_1, u_2$, we impose Lagrange's first constraint:

Then $y_p' = u_1 y_1' + u_2 y_2'$, and differentiating again:

Substituting $y_p, y_p', y_p''$ into $y'' + P y' + Q y = g(x)$:

Since $y_1$ and $y_2$ satisfy the homogeneous equation, the bracketed terms vanish, leaving:

3. Solution via Cramer's Rule

Constraints 1 and 2 form a linear algebraic system for the derivatives $u_1'$ and $u_2'$:

The coefficient matrix determinant is precisely the Wronskian $W(y_1, y_2)(x)$. By Cramer's Rule:

Integrating $u_1'$ and $u_2'$ yields the universal particular integral: