Variation of Parameters for Nonhomogeneous ODEs Cheat Sheet

Variation of parameters is a method for finding particular solutions to nonhomogeneous linear differential equations. It is especially useful when undetermined coefficients does not apply, such as equations with $\ln x$ , $\tan x$ , or variable coefficients. This cheat sheet helps students organize the setup, formulas, and solution steps for second-order and higher-order ODEs.

It is designed as a quick reference for homework, exams, and review.

Key Facts

For a second-order equation $y'' + p(x)y' + q(x)y = g(x)$ , first solve the homogeneous equation $y'' + p(x)y' + q(x)y = 0$ .
If $y_1$ and $y_2$ are linearly independent homogeneous solutions, then $y_h = C_1y_1 + C_2y_2$ .
The Wronskian for two solutions is $W(y_1,y_2) = y_1y_2' - y_1'y_2$ , and variation of parameters requires $W(y_1,y_2) \neq 0$ on the interval.
A particular solution has 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 found.
For $y'' + p(x)y' + q(x)y = g(x)$ , the derivative formulas are $u_1' = -\frac{y_2g}{W}$ and $u_2' = \frac{y_1g}{W}$ .
After finding $u_1'$ and $u_2'$ , integrate to get $u_1 = \int -\frac{y_2g}{W}\,dx$ and $u_2 = \int \frac{y_1g}{W}\,dx$ .
The general solution of a nonhomogeneous linear ODE is $y = y_h + y_p$ .
If the equation is not in standard form, divide by the leading coefficient so that the coefficient of $y''$ is $1$ before applying the formulas.

Vocabulary

Nonhomogeneous ODE: A differential equation with a nonzero forcing term, such as $y'' + p(x)y' + q(x)y = g(x)$ where $g(x) \neq 0$ .
Homogeneous equation: The related equation formed by setting the forcing term equal to zero, such as $y'' + p(x)y' + q(x)y = 0$ .
Fundamental solutions: A set of linearly independent solutions to the homogeneous equation that can be combined to form $y_h$ .
Wronskian: A determinant used to test linear independence, given by $W(y_1,y_2) = y_1y_2' - y_1'y_2$ for two functions.
Particular solution: One solution $y_p$ that satisfies the full nonhomogeneous differential equation.
General solution: The complete solution $y = y_h + y_p$ that combines the homogeneous solution with a particular solution.

Common Mistakes to Avoid

Forgetting to put the ODE in standard form is wrong because the formulas $u_1' = -\frac{y_2g}{W}$ and $u_2' = \frac{y_1g}{W}$ assume the coefficient of $y''$ is $1$ .
Using dependent homogeneous solutions is wrong because $W(y_1,y_2) = 0$ makes the variation of parameters formulas invalid.
Dropping the negative sign in $u_1' = -\frac{y_2g}{W}$ is wrong because it changes the particular solution and usually fails when substituted back into the ODE.
Adding arbitrary constants to $u_1$ and $u_2$ is unnecessary because those constants only reproduce terms already included in $y_h = C_1y_1 + C_2y_2$ .
Stopping after finding $u_1'$ and $u_2'$ is wrong because the functions $u_1$ and $u_2$ must be obtained by integration before building $y_p$ .

Practice Questions

1 Use variation of parameters to solve $y'' + y = \sec x$ on an interval where $\cos x \neq 0$ .
2 For $y'' - 3y' + 2y = e^{3x}$ , use $y_1 = e^x$ and $y_2 = e^{2x}$ to compute $W(y_1,y_2)$ , then set up $u_1'$ and $u_2'$ .
3 Find a particular solution for $y'' + 4y = \tan(2x)$ using $y_1 = \cos(2x)$ and $y_2 = \sin(2x)$ .
4 Explain why variation of parameters can work for forcing terms where undetermined coefficients does not apply.

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Variation of Parameters for Nonhomogeneous ODEs Cheat Sheet

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