ODE: Undetermined Coefficients for Systems

Solving nonhomogeneous systems of linear ordinary differential equations is fundamental to modeling driven engineering systems, from circuit networks to mechanical vibrations. While finding the homogeneous solution describes the system's natural behavior, the particular solution captures its forced response. The Method of Undetermined Coefficients provides a systematic, algebraic technique to find this particular solution when the system's forcing function is a combination of polynomials, exponentials, sines, and cosines. Mastering this method allows you to predict the complete system response, a critical skill for control systems, signal processing, and structural dynamics.

From Single Equations to Systems: The Core Analogy

The method for systems directly extends the logic used for single linear ODEs. For a single equation like $y^{\prime \prime }+p(t)y^{\prime }+q(t)y=g(t)$, if $g(t)$ is a function like $t^2$ or $e^{at}\cos (bt)$, you propose a trial solution with the same functional form but with undetermined constant coefficients. For a system of equations with constant coefficients, written in matrix form as ${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{g}(t)$, the principle is identical: you propose a trial vector solution ${\mathbf{x}}_p(t)$ whose components mirror the functional form of the forcing vector $\mathbf{g}(t)$.

The key difference is that the "undetermined coefficients" are now vectors of constants. If the forcing term in a 2x2 system is $\mathbf{g}(t)=\left[\begin{array}{c}{t}^2\\ 3t\end{array}\right]$, your trial particular solution isn't just a scalar polynomial; it's a vector of polynomials: ${\mathbf{x}}_p(t)=\left[\begin{array}{c}{a}_2{t}^2+a_{1}t+a_0\\ b_2{t}^2+b_{1}t+b_0\end{array}\right]$. You then substitute ${\mathbf{x}}_p(t)$ and its derivative ${\mathbf{x}}_p^{\prime }(t)$ into the system ${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{g}(t)$. This generates a system of algebraic equations by matching the coefficients of like terms (e.g., all $t^2$ terms, all $t$ terms, constants) across each row. Solving this algebraic system yields the numeric values for your vector of undetermined coefficients.

Structuring the Trial Solution for Different Forcing Types

The success of the method hinges on correctly choosing the initial form of ${\mathbf{x}}_p(t)$. The forcing vector $\mathbf{g}(t)$ dictates the form, component by component. You must consider each type of function present across all components of $\mathbf{g}(t)$.

• Polynomial Forcing: If any component of $\mathbf{g}(t)$ is a polynomial of degree $n$, then the corresponding component in ${\mathbf{x}}_p(t)$ must be a general polynomial of degree $n$, including all lower-order terms. For $\mathbf{g}(t)=\left[\begin{array}{c}5t\\ 2\end{array}\right]$, the trial is ${\mathbf{x}}_p(t)=\left[\begin{array}{c}{a}_{1}t+a_0\\ b_0\end{array}\right]$.
• Exponential Forcing: For a forcing term like $e^{rt}$, the trial solution includes a term of the form $\mathbf{v}{e}^{rt}$, where $\mathbf{v}$ is a constant vector to be determined. For $\mathbf{g}(t)=\left[\begin{array}{c}4e^{2t}\\ -e^{2t}\end{array}\right]$, the trial is ${\mathbf{x}}_p(t)=\left[\begin{array}{c}a\\ b\end{array}\right]e^{2t}$.
• Trigonometric Forcing: For forcing terms like $\sin (\omega t)$ or $\cos (\omega t)$, you must include both sine and cosine terms in your trial solution, even if only one appears in the forcing. This is because differentiation links sines and cosines. For $\mathbf{g}(t)=\left[\begin{array}{c}\sin (3t)\\ 0\end{array}\right]$, the correct trial is ${\mathbf{x}}_p(t)=\left[\begin{array}{c}a\cos (3t)+b\sin (3t)\\ c\cos (3t)+d\sin (3t)\end{array}\right]$.

Combined Forcing: When $\mathbf{g}(t)$ is a sum of these types, your trial ${\mathbf{x}}_p(t)$ is the sum of the corresponding trial vectors. For $\mathbf{g}(t)=\left[\begin{array}{c}t+e^{-t}\\ \cos (2t)\end{array}\right]$, you would propose: $${\mathbf{x}}_p(t)=\underset{\text{for }t}{\underbrace{\left[\begin{array}{c}{a}_{1}t+a_0\\ b_{1}t+b_0\end{array}\right]}}+\underset{\text{for }{e}^{-t}}{\underbrace{\left[\begin{array}{c}c\\ d\end{array}\right]e^{-t}}}+\underset{\text{for }\cos (2t)}{\underbrace{\left[\begin{array}{c}e\cos (2t)+f\sin (2t)\\ g\cos (2t)+h\sin (2t)\end{array}\right]}}.$$

The Critical Step: Handling Resonance (Modification Rule)

A major pitfall occurs when part of your proposed trial solution is already a solution to the associated homogeneous system ${\mathbf{x}}^{\prime }=A\mathbf{x}$. This is resonance, where the forcing function "matches" a natural mode of the system. In such cases, the standard trial solution will fail because it is not linearly independent from the homogeneous solution. You must modify it by multiplying the offending part of the trial by $t$ (or $t^s$, where $s$ is the smallest positive integer that removes the duplication).

How to check: Always solve for the homogeneous solution ${\mathbf{x}}_h(t)$ first. If any term in your proposed ${\mathbf{x}}_p(t)$ is a constant multiple of a term in ${\mathbf{x}}_h(t)$, resonance exists.

Example: Suppose for a system, ${\mathbf{x}}_h(t)=c_1\left[\begin{array}{c}1\\ -1\end{array}\right]e^t+c_2\left[\begin{array}{c}1\\ 2\end{array}\right]e^{-2t}$. If the forcing is $\mathbf{g}(t)=\left[\begin{array}{c}3e^t\\ -3e^t\end{array}\right]$, note that this forcing is a scalar multiple of the first homogeneous basis vector's function, $e^t$. A standard trial of ${\mathbf{x}}_p(t)=\left[\begin{array}{c}a\\ b\end{array}\right]e^t$ would be incorrect because $\left[\begin{array}{c}1\\ -1\end{array}\right]e^t$ is already in ${\mathbf{x}}_h(t)$. You must modify it: ${\mathbf{x}}_p(t)=t\left[\begin{array}{c}a\\ b\end{array}\right]e^t=\left[\begin{array}{c}at{e}^t\\ bt{e}^t\end{array}\right]$. Now substitute this modified form into the differential system to solve for $a$ and $b$.

Constructing the Complete System Response

Once you have successfully determined the particular solution ${\mathbf{x}}_p(t)$ and the homogeneous solution ${\mathbf{x}}_h(t)$, the general solution to the nonhomogeneous system is their sum: $$\mathbf{x}(t)={\mathbf{x}}_h(t)+{\mathbf{x}}_p(t).$$ The homogeneous solution contains arbitrary constants (e.g., $c_1,c_2$), while the particular solution is a single, specific vector function with no arbitrary constants. To solve an initial value problem, you would apply the initial conditions to this complete general solution $\mathbf{x}(t)$ to determine the values of the constants $c_1,c_2$, etc. This complete solution models the total system output: the transient response (homogeneous part, which often decays) plus the steady-state forced response (particular part).

Common Pitfalls

1. Insufficient Trial for Trigonometry: Proposing ${\mathbf{x}}_p(t)=\left[\begin{array}{c}a\\ b\end{array}\right]\sin (\omega t)$ for a forcing with $\sin (\omega t)$. This will always lead to an unsolvable system because the derivative produces $\cos (\omega t)$, which has no counterpart to match. Correction: Always include both sine and cosine terms with independent coefficient vectors for any trigonometric forcing.

1. Missing Lower-Order Terms in Polynomials: For forcing $\mathbf{g}(t)=\left[\begin{array}{c}{t}^2\\ 1\end{array}\right]$, proposing ${\mathbf{x}}_p(t)=\left[\begin{array}{c}a{t}^2\\ b\end{array}\right]$ is incorrect. The first component needs all lower-degree terms: $a_2{t}^2+a_{1}t+a_0$. Correction: For a polynomial of degree $n$, include terms for $t^n,t^{n-1},...,t,constant$ in the corresponding component.

1. Overlooking System-Wide Resonance: Focusing only on the functional form and not its vector relationship to the homogeneous solution. Resonance occurs if the entire term in the trial (function and its constant vector structure) solves the homogeneous system. Correction: After finding ${\mathbf{x}}_h(t)$, compare the functional form and the vector coefficient of each term. If a term in ${\mathbf{x}}_p(t)$ is a scalar multiple of a term in ${\mathbf{x}}_h(t)$, you must modify by multiplying by $t$.

1. Incorrect Substitution into Matrix Form: A procedural error when substituting ${\mathbf{x}}_p$ into ${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{g}$. Correction: Compute the derivative ${\mathbf{x}}_p^{\prime }$ correctly. Then compute the matrix product $A{\mathbf{x}}_p$. The equation ${\mathbf{x}}_p^{\prime }=A{\mathbf{x}}_p+\mathbf{g}(t)$ must hold for all $t$. Collect terms for each unique function ($e^{rt},t^n,\cos (\omega t),\sin (\omega t)$) to set up your algebraic system.

Summary

• The Method of Undetermined Coefficients for systems extends the scalar method: you propose a vector trial solution ${\mathbf{x}}_p(t)$ whose components mirror the functional forms present in the forcing vector $\mathbf{g}(t)$.
• The trial solution must account for all function types (polynomial, exponential, trigonometric) across all system components, and for trigonometric forcing, both sine and cosine terms must be included even if only one appears in the forcing.
• Resonance requires modification of the trial solution. If a term in your proposed ${\mathbf{x}}_p(t)$ is already present in the homogeneous solution ${\mathbf{x}}_h(t)$, multiply that entire term in the trial by $t$ (or a sufficient power of $t$) to achieve linear independence.
• The complete general solution is the sum $\mathbf{x}(t)={\mathbf{x}}_h(t)+{\mathbf{x}}_p(t)$, combining the system's natural and forced responses.
• The method is purely algebraic after the correct trial form is chosen. Success hinges on careful formulation of the trial, vigilant checking for resonance, and meticulous substitution into the matrix differential equation.