ODE: Variation of Parameters

When facing a nonhomogeneous linear differential equation—the kind that models everything from forced mechanical vibrations to current in an RLC circuit—finding a particular solution is the final step to a complete answer. While the method of Undetermined Coefficients works for specific, simple forcing functions, it fails for arbitrary ones like $e^x\sec (x)$ or $t^{-1}\ln (t)$. Enter the Variation of Parameters, a powerful, systematic algorithm that works for any continuous nonhomogeneous term, provided you first know the solution to the associated homogeneous equation. This method trades guesswork for computation, offering a reliable, if sometimes involved, path to the particular solution $y_p(t)$.

The Foundational Idea: From Constants to Functions

The logic of Variation of Parameters is an elegant extension of what you already know. Consider a general second-order linear ODE: $$y^{\prime \prime }+p(t)y^{\prime }+q(t)y=g(t).$$ Assume you have solved the corresponding homogeneous equation ($g(t)=0$) and found a fundamental set of solutions $\{y_1(t),y_2(t)\}$. The general homogeneous solution is $y_h(t)=c_1{y}_1(t)+c_2{y}_2(t)$, where $c_1$ and $c_2$ are constants.

The key insight is to vary these parameters—to promote the constants to unknown functions $u_1(t)$ and $u_2(t)$. We postulate that a particular solution to the nonhomogeneous equation can be written in the same form as the homogeneous solution, but with these variable functions: $$y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t).$$ If we simply substitute this into the ODE, we have one equation but two unknown functions ($u_1$ and $u_2$). To make the problem tractable, we impose a clever auxiliary condition to simplify the derivatives: we demand that $$u_1^{\prime }{y}_1+u_2^{\prime }{y}_2=0.$$ This condition is chosen to eliminate the terms involving second derivatives of $u_1$ and $u_2$ when we compute $y_p^{\prime \prime }$. With this condition, the first derivative of $y_p$ simplifies to: $$y_p^{\prime }=u_1{y}_1^{\prime }+u_2{y}_2^{\prime }.$$ Taking the derivative again (now the product rule will involve $u_1^{\prime }$ and $u_2^{\prime }$) and substituting $y_p$, $y_p^{\prime }$, and $y_p^{\prime \prime }$ into the original nonhomogeneous ODE leads, after significant simplification using the fact that $y_1$ and $y_2$ are homogeneous solutions, to a second condition: $$u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }=g(t).$$

The Role of the Wronskian

We now have a system of two linear equations for the derivatives $u_1^{\prime }$ and $u_2^{\prime }$: $$\{\begin{array}{l}{u}_1^{\prime }{y}_1+u_2^{\prime }{y}_2=0\\ u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }=g(t)\end{array}$$ This system can be solved using Cramer's rule. The determinant of the coefficient matrix is precisely the Wronskian, $W(y_1,y_2)(t)$, defined as: $$W(y_1,y_2)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }.$$ For a fundamental set of solutions, the Wronskian is always non-zero. Solving the system gives the formulas at the heart of the method: $$u_1^{\prime }(t)=\frac{-y_2(t)g(t)}{W(y_1,y_2)(t)},u_2^{\prime }(t)=\frac{y_1(t)g(t)}{W(y_1,y_2)(t)}.$$ The particular solution is then obtained by integration: $$u_1(t)=\int \frac{-y_2(t)g(t)}{W(t)}dt,u_2(t)=\int \frac{y_1(t)g(t)}{W(t)}dt,\text{and finally }{y}_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t).$$ Note that we use the indefinite integral here, and the constants of integration can be set to zero, as they would only add multiples of the homogeneous solution to $y_p$, which is already accounted for in $y_h$.

Extension to Higher-Order Equations

The method generalizes elegantly to an $n$th-order linear ODE of the form $$y^{(n)}+p_{n-1}(t)y^{(n-1)}+\cdots +p_1(t)y^{\prime }+p_0(t)y=g(t).$$ Given a fundamental set $\{y_1,y_2,\dots ,y_n\}$ for the homogeneous equation, we seek a particular solution as $$y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t)+\cdots +u_n(t)y_n(t).$$ We impose $n-1$ auxiliary conditions to streamline differentiation, all set to zero: $$\begin{array}{rl}{u}_1^{\prime }{y}_1+u_2^{\prime }{y}_2+\cdots +u_n^{\prime }{y}_n& =0,\\ u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }+\cdots +u_n^{\prime }{y}_n^{\prime }& =0,\\ & ⋮\\ u_1^{\prime }{y}_1^{(n-2)}+u_2^{\prime }{y}_2^{(n-2)}+\cdots +u_n^{\prime }{y}_n^{(n-2)}& =0.\end{array}$$ The final condition comes from substituting into the ODE itself: $$u_1^{\prime }{y}_1^{(n-1)}+u_2^{\prime }{y}_2^{(n-1)}+\cdots +u_n^{\prime }{y}_n^{(n-1)}=g(t).$$ This system is solved for $u_1^{\prime },u_2^{\prime },\dots ,u_n^{\prime }$ using the Wronskian of the fundamental set, $W(y_1,y_2,\dots ,y_n)(t)$. The formula for each is analogous, involving cofactor determinants. The process is computationally heavier but remains a guaranteed procedure.

Advantages Over Undetermined Coefficients

Variation of Parameters holds two decisive advantages that make it indispensable for engineering analysis:

1. Universal Applicability: It works for any continuous nonhomogeneous term $g(t)$. Undetermined Coefficients is limited to $g(t)$ that are a sum of terms of the form $t^m{e}^{\alpha t}\cos (\beta t)$ or $t^m{e}^{\alpha t}\sin (\beta t)$. If $g(t)$ is $\ln (t)$, $\sec (t)$, or a non-standard function, Variation of Parameters is your only direct analytical choice.
2. Theoretical Clarity: It provides a closed-form integral formula for the particular solution, which is useful for theoretical proofs and sensitivity analysis. It directly reveals how the forcing function $g(t)$ is "filtered" through the homogeneous solutions.

The trade-off is computational complexity. The integrals you get can be non-elementary or messy, and the method requires you to fully solve the homogeneous equation first. Undetermined Coefficients, when applicable, is usually faster and involves simpler algebra.

Computational Procedure: A Step-by-Step Guide

Let's solidify the method with a structured workflow for a second-order equation:

1. Solve the Homogeneous ODE: Find a fundamental set of solutions $\{y_1,y_2\}$ and the homogeneous solution $y_h=c_1{y}_1+c_2{y}_2$.
2. Compute the Wronskian: Calculate $W(t)=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$. Simplify it. This step is critical; an error here will propagate.
3. Set Up the Integrands: Form the expressions for the derivatives of the parameter functions:

$$u_1^{\prime }=\frac{-y_2\cdot g(t)}{W(t)},u_2^{\prime }=\frac{y_1\cdot g(t)}{W(t)}.$$

1. Integrate: Find $u_1(t)$ and $u_2(t)$ by integrating the expressions from the previous step:

$$u_1(t)=\int u_1^{\prime }(t)dt,u_2(t)=\int u_2^{\prime }(t)dt.$$ Omit the constants of integration.

1. Construct the Particular Solution: Assemble $y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t)$.
2. Form the General Solution: The complete solution to the nonhomogeneous ODE is $y(t)=y_h(t)+y_p(t)$.

Worked Example: Find a particular solution to $y^{\prime \prime }+y=\sec (t)$.

• Step 1: The homogeneous solution is $y_h=c_1\cos (t)+c_2\sin (t)$. So, $y_1=\cos (t)$, $y_2=\sin (t)$.
• Step 2: Compute the Wronskian: $W=(\cos (t))(\cos (t))-(\sin (t))(-\sin (t))={\cos }^2(t)+{\sin }^2(t)=1$.
• Step 3: Set up integrands with $g(t)=\sec (t)$:

$$u_1^{\prime }=\frac{-\sin (t)\sec (t)}{1}=-\tan (t),u_2^{\prime }=\frac{\cos (t)\sec (t)}{1}=1.$$

• Step 4: Integrate:

$$u_1=\int -\tan (t)dt=\ln |\cos (t)|,u_2=\int 1dt=t.$$

• Step 5 & 6: Construct $y_p$ and the general solution:

$$y_p=\ln |\cos (t)|\cdot \cos (t)+t\cdot \sin (t),$$ $$y=c_1\cos (t)+c_2\sin (t)+\cos (t)\ln |\cos (t)|+t\sin (t).$$

Common Pitfalls

1. Applying the Method Before Solving the Homogeneous Equation: Variation of Parameters is not a guesswork method. You must have a correct fundamental set $\{y_1,y_2,...\}$ before you begin. Attempting to skip this step will lead to an unsolvable system.
2. Incorrect Wronskian Calculation: A sign error or simplification mistake in computing $W(t)$ will make the formulas for $u_1^{\prime }$ and $u_2^{\prime }$ incorrect. Always double-check this determinant. Remember, for a fundamental set, $W(t)\ne 0$.
3. Including Constants of Integration: When integrating $u_1^{\prime }$ and $u_2^{\prime }$, do not add constants $+C$. Adding them would yield $y_p=u_1{y}_1+u_2{y}_2+C_1{y}_1+C_2{y}_2$, where the last two terms are just part of the homogeneous solution. The method is designed to produce a particular solution without these extra terms.
4. Misidentifying $g(t)$: Ensure the ODE is in standard form (leading coefficient 1) before identifying $g(t)$. For an equation like $t^2{y}^{\prime \prime }+t{y}^{\prime }+y=\ln (t)$, you must first divide by $t^2$ to get $y^{\prime \prime }+\frac{1}{t}{y}^{\prime }+\frac{1}{t^2}y=\frac{\ln (t)}{t^2}$. Here, $g(t)=\frac{\ln (t)}{t^2}$, not $\ln (t)$.

Summary

• Variation of Parameters is a general method for finding a particular solution $y_p$ to a nonhomogeneous linear ODE by replacing the constants in the homogeneous solution with unknown functions and solving for them.
• The method's core machinery relies on the Wronskian of the fundamental solution set to derive formulas for the derivatives of these variable parameters.
• It is applicable to ODEs of any order and, crucially, works for arbitrary continuous forcing functions $g(t)$, making it more versatile than the method of Undetermined Coefficients.
• The computational procedure is systematic: 1) Solve the homogeneous ODE, 2) Compute the Wronskian, 3) Form the integrands for $u_i^{\prime }$, 4) Integrate to find $u_i$, 5) Assemble $y_p$.
• The primary trade-off for its generality is increased computational complexity, often involving integrals that are difficult to evaluate, but the method guarantees a correct analytical path forward where other methods fail.