Chapter 2 · Second-Order ODEs

Variation of Parameters

The Universal Method for ANY Forcing Function

Dr. Mohamed Mabrok · Qatar University

Why We Need Variation of Parameters

Undetermined Coefficients
(Limited)

Polynomials, exponentials, trig

$ay'' + by' + cy = g(t)$

Variation of Parameters
(Universal!)

Works for ANY g(t)

When UC Fails

      Functions like $\tan(t)$, $\sec(t)$, $\ln(t)$, $1/t$, $e^t/t^2$, etc. require Variation of Parameters!

The Brilliant Idea: Varying the Parameters

Standard Homogeneous Solution

      $$y_c = C_1 y_1(t) + C_2 y_2(t)$$

The Key Insight

      What if the constants aren't constant?
    
      $$C_1 \to u_1(t), \quad C_2 \to u_2(t)$$

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

Why This Works

      By letting $C_1$ and $C_2$ vary as functions, we gain enough flexibility to match ANY forcing function $g(t)$.

Deriving the Variation of Parameters Formulas

STEP 1 Start with $y_p = u_1 y_1 + u_2 y_2$ and compute derivatives

$$y_p' = u_1' y_1 + u_1 y_1' + u_2' y_2 + u_2 y_2'$$

STEP 2 Impose constraint: $u_1' y_1 + u_2' y_2 = 0$ (simplification)

$$y_p' = u_1 y_1' + u_2 y_2'$$

STEP 3 Compute $y_p''$

$$y_p'' = u_1' y_1' + u_1 y_1'' + u_2' y_2' + u_2 y_2''$$

STEP 4 Substitute into $ay'' + by' + cy = g(t)$ and simplify

u_1' y_1' + u_2' y_2' = \frac{g(t)}{a}

The Wronskian Determinant

We now have a system of two equations:

u_1' y_1 + u_2' y_2 = 0$$ $$u_1' y_1' + u_2' y_2' = \frac{g(t)}{a}

Apply Cramer's Rule

      Solving this system introduces a key determinant:
    
      $$W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$$

Fundamental Property

      If $W(y_1, y_2) \neq 0$, then $y_1$ and $y_2$ are linearly independent, and a unique particular solution exists!

The Variation of Parameters Formulas

The Complete Solution

      $$u_1(t) = -\int \frac{y_2(t) \cdot g(t)}{a \cdot W(t)} dt$$
    
      $$u_2(t) = \int \frac{y_1(t) \cdot g(t)}{a \cdot W(t)} dt$$
    
      $$y_p = u_1(t)y_1(t) + u_2(t)y_2(t)$$
    
      $$y = C_1 y_1 + C_2 y_2 + u_1 y_1 + u_2 y_2$$

Important!

      Don't forget the factor $1/a$ in the formulas! This is the coefficient of $y''$.

The 5-Step Method

1 Solve the homogeneous equation to find $y_1(t)$ and $y_2(t)$

2 Compute the Wronskian: $W = y_1 y_2' - y_2 y_1'$

3 Find $u_1(t) = -\int \frac{y_2 g}{aW} dt$

4 Find $u_2(t) = \int \frac{y_1 g}{aW} dt$

5 Assemble: $y = C_1 y_1 + C_2 y_2 + u_1 y_1 + u_2 y_2$

Pro Tip

      Steps 3 and 4 (the integrations) are often the hardest part. Take your time with these!

Worked Example

Solve: $y'' + y = \tan(t)$

STEPS 1-2 Homogeneous solution and Wronskian

r^2 + 1 = 0 \implies y_1 = \cos(t), \quad y_2 = \sin(t)$$ $$W = \cos^2(t) + \sin^2(t) = 1

STEPS 3-4 Find $u_1$ and $u_2$

u_1 = -\int \sin(t) \tan(t) dt = \sin(t) - \ln|\sec(t) + \tan(t)|$$ $$u_2 = \int \cos(t) \tan(t) dt = -\cos(t)

STEP 5 General solution

y = C_1 \cos(t) + C_2 \sin(t) - \cos(t) \ln|\sec(t) + \tan(t)|

Worked Example

Solve: $y'' + 9y = 9\sec^2(3t)$

STEPS 1-2 Setup

r^2 + 9 = 0 \implies y_1 = \cos(3t), \quad y_2 = \sin(3t)$$ $$W = 3

STEPS 3-4 Integrate

u_1 = -\int \frac{\sin(3t) \cdot 9\sec^2(3t)}{3} dt = -\sec(3t)$$ $$u_2 = \int \frac{\cos(3t) \cdot 9\sec^2(3t)}{3} dt = \ln|\sec(3t) + \tan(3t)|

STEP 5 Solution

y = C_1\cos(3t) + C_2\sin(3t) - 1 + \sin(3t)\ln|\sec(3t) + \tan(3t)|

Worked Example

Solve: $y'' - y = e^t \sin(t)$

STEPS 1-2 Characteristic equation

r^2 - 1 = 0 \implies r = \pm 1$$ $$y_1 = e^t, \quad y_2 = e^{-t}, \quad W = -2

STEPS 3-4 Integrals

u_1 = \frac{1}{2}\int \sin(t) dt = -\frac{1}{2}\cos(t)$$ $$u_2 = -\frac{1}{2}\int e^{2t}\sin(t) dt = -\frac{e^{2t}(2\sin(t) + \cos(t))}{10}

STEP 5 General solution

y = C_1 e^t + C_2 e^{-t} - e^t\frac{2\cos(t) + \sin(t)}{5}

Worked Example

Solve: $y'' + 2y' + y = t^2 e^{-t}$

STEPS 1-2 Repeated roots

(r+1)^2 = 0 \implies y_1 = e^{-t}, \quad y_2 = t e^{-t}, \quad W = e^{-2t}

STEPS 3-4 Find $u_1$ and $u_2$

u_1 = -\int t^3 dt = -\frac{t^4}{4}$$ $$u_2 = \int t^2 dt = \frac{t^3}{3}

STEP 5 Particular solution

y = C_1 e^{-t} + C_2 t e^{-t} + \frac{t^4 e^{-t}}{12}

Comparison: Undetermined Coefficients vs Variation of Parameters

Feature	Undetermined Coefficients	Variation of Parameters
Works for which g(t)?	Polynomials, exponentials, sin/cos	ANY function
Method Complexity	Easy (guess & match)	Requires integration
Resonance	Needs modification	Handles naturally
Integration	No	Yes (often hard)
Wronskian Needed	No	Yes

Use UC When...

        $g(t)$ is polynomial, exponential, or trig. UC is faster when applicable!

Use VP When...

        $g(t)$ is exotic ($\tan$, $\sec$, $\ln$, $1/t$, etc.) or UC doesn't apply.

Worked Example

Solve: $y'' - 2y' + y = \frac{e^t}{t^2}$

UC cannot handle $1/t^2$. VP is the only method!

STEPS 1-2 Homogeneous solution

(r-1)^2 = 0 \implies y_1 = e^t, \quad y_2 = t e^t, \quad W = e^{2t}

STEPS 3-4 Integration

u_1 = -\int \frac{1}{t} dt = -\ln|t|$$ $$u_2 = \int \frac{1}{t^2} dt = -\frac{1}{t}

STEP 5 Solution

y = C_1 e^t + C_2 t e^t - e^t(\ln|t| + 1)

Common Mistakes to Avoid

⚠️ Typical Errors

Forgetting the factor $1/a$: Always divide $g(t)$ by the leading coefficient!
Wronskian computation: Watch for sign errors in $y_1 y_2' - y_2 y_1'$.
Missing the negative sign: Remember $u_1 = -\int \frac{y_2 g}{aW} dt$ (with minus!).
Not simplifying $y_p$: Terms often cancel after algebra — always expand and simplify.
Using VP unnecessarily: If UC works, use it — VP is a last resort for exotic functions.

Key Verification

      Always verify $W \neq 0$ before proceeding. If $W = 0$, your solution functions are linearly dependent!

Key Takeaways & Summary

The Variation of Parameters Method

The Core Idea: Replace constants $C_1, C_2$ with functions $u_1(t), u_2(t)$.
The Constraint: Impose $u_1' y_1 + u_2' y_2 = 0$ to simplify the algebra.
The Wronskian: Compute $W = y_1 y_2' - y_2 y_1'$ — this is essential!
The Formulas: Use $u_1 = -\int \frac{y_2 g}{aW} dt$ and $u_2 = \int \frac{y_1 g}{aW} dt$.
Universal Power: VP works for ANY $g(t)$ — this is your ultimate weapon!

Next Topic

      Coming up: Euler Equations — how to handle ODEs with variable coefficients, not just constants.