Variation of Parameters

Precise definition

For normalized $y''+p(x)y'+q(x)y=g(x)$ with fundamental homogeneous solutions $y_1,y_2$ and Wronskian $W=y_1y_2'-y_1'y_2\ne0$, variation of parameters seeks $y_p=u_1y_1+u_2y_2$ where $u_1'=-y_2g/W$ and $u_2'=y_1g/W$.

Notation and mathematical language

The auxiliary condition $u_1'y_1+u_2'y_2=0$ eliminates second derivatives of the unknown parameter functions. If the original leading coefficient is $a(x)$, divide first; otherwise the $g$ in the formulas is wrong. Constants arising in $u_1,u_2$ can be set to zero because they add homogeneous terms.

Conceptual picture

The method lets the coefficients of homogeneous modes vary just enough to absorb forcing. The Wronskian denominator solves a two-by-two system for $u_1',u_2'$, so its nonzero condition is the same independence requirement needed for a fundamental set.

Conditions and key results

Coefficient continuity and $W\ne0$ are required on the interval. The method may yield nonelementary integrals, which still define an exact particular solution. Signs depend on the Wronskian convention, so formulas and $W$ order must be consistent.

A reliable strategy

1. Normalize the equation and find a verified fundamental homogeneous pair.
2. Compute $W=y_1y_2'-y_1'y_2$ and ensure it is nonzero on the interval.
3. Calculate $u_1',u_2'$, integrate, and form $y_p=u_1y_1+u_2y_2$.
4. Add $y_h$, apply data if present, and substitute $y_p$ into the nonhomogeneous operator.

Fully worked example

Interpretation and application

Variation of parameters handles forcing outside the undetermined-coefficients catalogue and variable coefficients when a homogeneous basis is known. Exact integral expressions may be more informative than unstable decimal antiderivatives and should be distinguished from numerical approximations.

Common mistakes

Verification and reasonableness

• Differentiate $y_p$ and substitute into the normalized equation.
• Confirm the chosen interval keeps coefficients, forcing, and Wronskian valid.
• Compare with an undetermined-coefficients answer when both methods apply; their difference should be homogeneous.

Practice

1. What must be true of $W$?
2. Why normalize first?
3. What happens to constants in $u_1,u_2$?

Further deduction

A definite-integral version builds in a reference point $x_0$: $u_1(x)=-\int_{x_0}^x y_2(s)g(s)/W(s)\,ds$ and $u_2(x)=\int_{x_0}^x y_1(s)g(s)/W(s)\,ds$. It avoids arbitrary constants and produces a particular solution satisfying $y_p(x_0)=y_p'(x_0)=0$, which is convenient when initial data are applied separately to the homogeneous constants.

The two auxiliary equations are $u_1'y_1+u_2'y_2=0$ and $u_1'y_1'+u_2'y_2'=g$. Solving this system by Cramer's rule produces the stated Wronskian formulas. Deriving them once makes the signs recoverable and shows why a zero Wronskian blocks the method: the homogeneous pair does not provide independent coordinate directions for the forced adjustment.

Abel's identity gives $W(x)=W(x_0)e^{-\int_{x_0}^xP(s)ds}$ for the normalized homogeneous equation. Thus a Wronskian nonzero at one point stays nonzero throughout the coefficient interval, validating the denominator in variation-of-parameters formulas there.

Related topics

• Second-Order Linear Differential Equations
• Undetermined Coefficients
• Linear Independence