Method of Undetermined Coefficients

The method of undetermined coefficients provides a systematic technique for finding particular solutions to nonhomogeneous linear differential equations when the forcing function has a specific form.

DefinitionNonhomogeneous Second-Order Linear ODE

A nonhomogeneous second-order linear differential equation has the form:

$ay'' + by' + cy = f(x)$

where $f(x) \neq 0$ is the forcing function or nonhomogeneous term. The general solution is:

$y = y_h + y_p$

where $y_h$ is the general solution of the homogeneous equation and $y_p$ is any particular solution.

Applicable Forms

The method works when $f(x)$ is a linear combination of:

Polynomials: $x^n$
Exponentials: $e^{ax}$
Sines and cosines: $\sin(bx)$ , $\cos(bx)$
Products of the above

Basic Strategy

The particular solution $y_p$ is guessed to have the same form as $f(x)$ (and its derivatives), with undetermined coefficients. Substitute into the equation and solve for the coefficients.

ExamplePolynomial Forcing

Solve $y'' - 3y' + 2y = 4x^2$ .

First find $y_h$ : Characteristic equation $r^2 - 3r + 2 = 0$ gives $(r-1)(r-2) = 0$ , so:

$y_h = c_1e^x + c_2e^{2x}$

For $y_p$ , since $f(x) = 4x^2$ is a polynomial of degree 2, try:

$y_p = Ax^2 + Bx + C$

Then $y_p' = 2Ax + B$ and $y_p'' = 2A$ .

Substituting:

$2A - 3(2Ax + B) + 2(Ax^2 + Bx + C) = 4x^2$

$2Ax^2 + (-6A + 2B)x + (2A - 3B + 2C) = 4x^2$

Comparing coefficients:

$x^2$ : $2A = 4 \Rightarrow A = 2$
$x^1$ : $-6A + 2B = 0 \Rightarrow B = 6$
$x^0$ : $2A - 3B + 2C = 0 \Rightarrow C = 7$

So $y_p = 2x^2 + 6x + 7$ .

General solution: $y = c_1e^x + c_2e^{2x} + 2x^2 + 6x + 7$

Remark

Modification Rule (Resonance): If any term in the guessed $y_p$ is a solution of the homogeneous equation, multiply the guess by $x$ (or $x^2$ if needed) until there's no overlap. This situation is called resonance.

ExampleExponential Forcing with Resonance

Solve $y'' - 3y' + 2y = 5e^{2x}$ .

The homogeneous solution is $y_h = c_1e^x + c_2e^{2x}$ .

Normally we'd try $y_p = Ae^{2x}$ , but $e^{2x}$ is already in $y_h$ (resonance!).

Apply modification rule: try $y_p = Axe^{2x}$ .

Then $y_p' = Ae^{2x} + 2Axe^{2x}$ and $y_p'' = 4Ae^{2x} + 4Axe^{2x}$ .

Substituting:

$(4Ae^{2x} + 4Axe^{2x}) - 3(Ae^{2x} + 2Axe^{2x}) + 2(Axe^{2x}) = 5e^{2x}$

$Ae^{2x} = 5e^{2x}$

Therefore $A = 5$ and $y_p = 5xe^{2x}$ .

General solution: $y = c_1e^x + c_2e^{2x} + 5xe^{2x}$

ExampleTrigonometric Forcing

Solve $y'' + 4y = 3\sin 2x$ .

Homogeneous solution: $r^2 + 4 = 0$ gives $r = \pm 2i$ , so:

$y_h = c_1\cos 2x + c_2\sin 2x$

Resonance occurs! Both $\sin 2x$ and $\cos 2x$ are in $y_h$ .

Try $y_p = x(A\cos 2x + B\sin 2x)$ .

After differentiation and substitution (details omitted):

$y_p = -\frac{3}{4}x\cos 2x$

General solution: $y = c_1\cos 2x + c_2\sin 2x - \frac{3}{4}x\cos 2x$

The method of undetermined coefficients is efficient for constant-coefficient equations with special forcing functions, but does not work for arbitrary $f(x)$ . For general cases, variation of parameters is needed.