Nonhomogeneous Equations: Undetermined Coefficients

Adding a forcing term to the homogeneous equation gives the nonhomogeneous equation

L [y] = y^{''} + p (t) y^{'} + q (t) y = g (t), g \neq \equiv 0. (N)

Physically $g$ is an external input: a driving force on a spring, an applied voltage on a circuit. The problem splits into two parts. One is the general solution of the homogeneous version. The other is a single particular solution of (N), and for a useful class of forcing functions it comes from an educated guess — the method of undetermined coefficients — together with one correction for when the guess collides with the homogeneous solution.

The structure of the general solution

Two solutions of (N) differ by a solution of the homogeneous equation, because $L$ is linear: $L [Y_{1} - Y_{2}] = L [Y_{1}] - L [Y_{2}] = g - g = 0$ . That observation determines the entire solution set.

Fix one particular solution $Y$ . Any other solution $ϕ$ satisfies $ϕ - Y = c_{1} y_{1} + c_{2} y_{2}$ , so $ϕ = c_{1} y_{1} + c_{2} y_{2} + Y$ . This is the general solution.

The two pieces have names.

Complementary solution $y_{c} = c_{1} y_{1} + c_{2} y_{2}$ : the general homogeneous solution, carrying both arbitrary constants and therefore all the freedom to meet initial conditions.
Particular solution $Y$ : one specific solution of the full equation (N), with no free constants.

Solving (N) therefore takes three steps: find $y_{c}$ , find any $Y$ , and add them. Initial conditions are applied only at the very end, to the sum, because $y_{c}$ carries the adjustable constants. Applying them to $Y$ alone or to $y_{c}$ alone is a common error.

The general nonhomogeneous solution is the complementary solution (all of the homogeneous freedom) plus one particular solution of the forced equation.

The trial-form guess

The method of undetermined coefficients finds $Y$ by assuming it has the same functional shape as $g$ , with unknown coefficients, then substituting to pin the coefficients down. It works because the derivatives of exponentials, sines, cosines, and polynomials stay within the same small families, so $L [Y]$ has the same shape as $Y$ and matching it to $g$ gives linear equations for the unknowns. The method is limited to constant-coefficient equations and to forcing terms of these types, but that covers a large share of applications.

Differentiation cycles sine and cosine into each other (up to sign), so a trigonometric trial must carry both terms to close under

L

Closure under differentiation determines every trial form. A polynomial of degree $n$ needs a full degree- $n$ trial $A_{0} t^{n} + \dots + A_{n}$ (all lower terms, since differentiation feeds them back), and a product such as $e^{α t} cos (β t)$ needs both $e^{α t} cos (β t)$ and $e^{α t} sin (β t)$ .

Each forcing type maps to a trial solution closed under differentiation, with

a, b

real constants; the coefficients

A, B

are solved for by substitution.

Sums and products of forcing terms

If the forcing is a sum $g = g_{1} + g_{2}$ , solve one equation per term and add the particular solutions. This follows from linearity: if $L [Y_{1}] = g_{1}$ and $L [Y_{2}] = g_{2}$ , then $L [Y_{1} + Y_{2}] = g_{1} + g_{2}$ .

A product of types, by contrast, stays together: $g = t e^{α t}$ is one term needing the trial $(A_{0} t + A_{1}) e^{α t}$ , not a sum of separate guesses. A product with a trigonometric factor needs both the sine and cosine versions.

Resonance and the multiplicity correction

The method has one failure mode, and it is physically important.

This is resonance: when the forcing frequency matches a natural mode of the system, the response grows with $t$ instead of staying bounded, and the trial solution must carry an explicit factor of $t$ to reflect it. The general rule:

The correction tracks the multiplicity of the offending root. If $e^{α t}$ is a simple root of the characteristic equation, one factor of $t$ suffices; if it is a double root (so both $e^{α t}$ and $t e^{α t}$ solve the homogeneous equation), two are needed.

The naive trial

A e^{- t}

lies in the homogeneous solution space, so

L

sends it to zero; multiplying by

t

moves it out and produces a genuine response.

When the complementary and particular parts are plotted together, the resonant particular solution is visibly the growing one: the complementary exponential decays while $t e^{- t}$ first rises.

For the resonant equation, the complementary part decays while the particular part

t e^{- t}

grows before decaying; their sum is the full response.

Trial-solution reference

The table collects the standard trials, before any multiplicity correction. $P_{n}$ denotes a general polynomial of degree $n$ , and $\tilde{P}_{n}$ the general polynomial trial $A_{0} t^{n} + \dots + A_{n}$ of the same degree.

Forcing term $g (t)$	Trial solution $Y (t)$
$P_{n} (t)$	$\tilde{P}_{n} (t)$
$e^{α t}$	$A e^{α t}$
$P_{n} (t) e^{α t}$	$\tilde{P}_{n} (t) e^{α t}$
$sin β t$ or $cos β t$	$A sin β t + B cos β t$
$e^{α t} sin β t$ or $e^{α t} cos β t$	$e^{α t} (A sin β t + B cos β t)$
$P_{n} (t) e^{α t} sin β t$	$e^{α t} (\tilde{P}_{n} sin β t + \tilde{Q}_{n} cos β t)$

In every row: if a term of the trial already solves the homogeneous equation, multiply the row's trial by $t$ (or $t^{2}$ for a double root) before solving for the coefficients.

The method is fast when the trial form is easy to write down, but it fails when $g$ falls outside these families, for instance $g (t) = tan t$ or $g (t) = 1/ t$ . The variation of parameters method removes that restriction at the cost of evaluating two integrals.¹²

Boyce, Elementary Differential Equations, §3.5 — Nonhomogeneous Equations; Method of Undetermined Coefficients: the complementary-plus-particular structure (Theorems 3.5.1–3.5.2), the trial-form guesses for polynomial/exponential/trigonometric forcing, sums of forcing terms, and the multiply-by- $t$ resonance correction. ↩
Simmons, Differential Equations with Applications and Historical Notes, §18 — The Method of Undetermined Coefficients: assuming a particular solution of the same form as the forcing term and determining its coefficients by substitution. ↩