Variation of Parameters

Variation of parameters is a general method for finding particular solutions to nonhomogeneous linear differential equations. Unlike undetermined coefficients, it works for any continuous forcing function.

DefinitionVariation of Parameters Method

Given the nonhomogeneous equation:

$y'' + p(x)y' + q(x)y = f(x)$

with homogeneous solutions $y_1$ and $y_2$ , the method seeks a particular solution of the form:

$y_p = u_1(x)y_1(x) + u_2(x)y_2(x)$

where $u_1$ and $u_2$ are functions to be determined (the "parameters" that "vary").

Derivation of the Method

The idea is to replace the constants in $y_h = c_1y_1 + c_2y_2$ with functions $u_1(x)$ and $u_2(x)$ .

Imposing the constraint:

$u_1'y_1 + u_2'y_2 = 0$

and requiring $y_p$ to satisfy the differential equation yields:

$u_1'y_1' + u_2'y_2' = f(x)$

These two equations can be solved for $u_1'$ and $u_2'$ using Cramer's rule.

Formula for Particular Solution

For the standard form $y'' + p(x)y' + q(x)y = f(x)$ :

$u_1' = -\frac{y_2f}{W}, \quad u_2' = \frac{y_1f}{W}$

where $W = W(y_1, y_2) = y_1y_2' - y_1'y_2$ is the Wronskian.

Integrating:

$y_p = -y_1\int\frac{y_2f}{W}dx + y_2\int\frac{y_1f}{W}dx$

ExampleUsing Variation of Parameters

Solve $y'' + y = \sec x$ .

Homogeneous solution: $r^2 + 1 = 0$ gives $y_1 = \cos x$ , $y_2 = \sin x$ .

Wronskian:

$W = \cos x \cdot \cos x - (-\sin x) \cdot \sin x = \cos^2 x + \sin^2 x = 1$

Using the formulas with $f(x) = \sec x$ :

$u_1' = -\frac{\sin x \cdot \sec x}{1} = -\tan x$

$u_2' = \frac{\cos x \cdot \sec x}{1} = 1$

Integrating:

$u_1 = \ln|\cos x|, \quad u_2 = x$

Therefore:

$y_p = \cos x \ln|\cos x| + x\sin x$

General solution: $y = c_1\cos x + c_2\sin x + \cos x \ln|\cos x| + x\sin x$

Remark

The variation of parameters method always works for any continuous $f(x)$ , but the required integrals may not have closed forms. In contrast, undetermined coefficients is limited to special forms of $f(x)$ but is often computationally simpler.

ExampleComparison with Undetermined Coefficients

Consider $y'' - 3y' + 2y = e^x$ .

Method 1 - Variation of Parameters:

Homogeneous solutions: $y_1 = e^x$ , $y_2 = e^{2x}$

Wronskian: $W = e^x \cdot 2e^{2x} - e^x \cdot e^{2x} = e^{3x}$

$u_1' = -\frac{e^{2x} \cdot e^x}{e^{3x}} = -1, \quad u_2' = \frac{e^x \cdot e^x}{e^{3x}} = e^{-x}$

$u_1 = -x, \quad u_2 = -e^{-x}$

$y_p = -xe^x - e^{2x} \cdot e^{-x} = -xe^x - e^x = -(x+1)e^x$

Method 2 - Undetermined Coefficients (with resonance):

Since $e^x$ is in $y_h$ , try $y_p = Axe^x$ .

Substituting gives $A = 1$ , so $y_p = xe^x$ .

(Note: We can drop the $-e^x$ term as it's part of $y_h$ .)

Both methods work, but undetermined coefficients is simpler here.

Remark

The Wronskian $W(y_1, y_2)$ is never zero when $y_1$ and $y_2$ are linearly independent solutions. This guarantees that the formulas for $u_1'$ and $u_2'$ are well-defined.

Variation of parameters is the most general method for solving nonhomogeneous linear ODEs and provides theoretical insight into the structure of solutions.