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.

Understanding Variation of Parameters for Nonhomogeneous ODEs

The main idea is to loosen the constants in a homogeneous solution. A homogeneous solution uses fixed numbers multiplying the basic solution shapes. For a forced equation, fixed numbers are too limited because the input can keep changing with x.

Variation of parameters allows those multipliers to change with x. This creates enough flexibility for the solution to respond to the forcing term. The method introduces two unknown functions, but it adds a useful simplifying condition.

The condition says that the first changing multiplier times the first basic solution, plus the second changing multiplier times the second basic solution, has derivative zero. This removes unwanted second derivatives of the new functions and turns the work into a manageable two equation system.

The Wronskian measures whether the two homogeneous solutions truly provide different directions for building solutions. If it is zero, the two solution shapes are dependent at that point or throughout the interval. Then the system used to find the changing multipliers cannot be solved uniquely.

This is similar to trying to locate a point on a map using two identical directions. Independent directions are needed. Students should pay close attention to the interval being used.

Coefficients such as one over x or tangent x can be undefined at certain points. A valid pair of fundamental solutions on one interval may not work across a point where the equation itself is undefined.

The integrals in this method often look harder than the differential equation. That is normal. The method is valuable because it reduces the differential equation problem to integration problems.

Sometimes those integrals have elementary answers. Sometimes they involve special functions or must be left as integrals. Leaving a particular solution in integral form can still be correct when the integrals cannot be simplified with methods from a standard calculus course.

Any constants produced during these integrations do not need to be kept in the particular solution. They only recreate terms already included in the homogeneous part. Choosing convenient lower limits for definite integrals is one clean way to avoid carrying extra constants.

This method appears whenever a system has its own natural behavior but is acted on by an external input. A spring can vibrate naturally, then respond to a changing applied force. An electrical circuit can have its own voltage patterns, then be driven by a source signal.

Population or chemical models can receive a time dependent input. In each case, the homogeneous solutions describe the system's built in response, while the parameter functions track how the input adds to it over time.

After completing the calculation, check the answer by differentiating the particular solution and substituting it into the original equation. Common errors include forgetting to first make the leading coefficient equal to one, using the Wronskian with reversed signs, and mixing up the order of the two fundamental solutions.

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Variation of Parameters for Nonhomogeneous ODEs Cheat Sheet

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Understanding Variation of Parameters for Nonhomogeneous ODEs