Chapter 2 · Second-Order ODEs

Method of Undetermined Coefficients

Finding Particular Solutions by Intelligent Guessing

Dr. Mohamed Mabrok · Qatar University

The Big Picture: Non-Homogeneous Equations

y_c (Complementary)

Solved in Ch2.1
(Homogeneous)

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

y_p (Particular)

THIS LECTURE
(For specific g(t))

Key Principle

      y = y_c + y_p
    
      General solution = Complementary (homogeneous) + Particular (non-homogeneous)

The Strategy: Guess Smartly!

1

STEP 1: Solve Homogeneous

Find $y_c$ by solving $ay'' + by' + cy = 0$ (Ch2.1 method)

y_c = C_1e^{r_1 t} + C_2e^{r_2 t} \text{ (or appropriate form)}

2

STEP 2: Guess the Form

Based on the form of g(t), write down a trial solution $y_p$

The key: Choose the right TYPE of function (polynomial, exponential, sine/cosine, etc.)

3

STEP 3: Match Coefficients

Substitute $y_p$ into the ODE, match coefficients, solve for unknowns

This gives you the specific constants in your particular solution

When Does This Method Work?

      Polynomials, exponentials, sines, cosines, and combinations thereof. NOT for functions like $\tan(t)$, $\ln(t)$, or $1/t$.

The Guessing Table: Your Reference Card

If g(t) looks like...	Then guess y_p as...	Note
Polynomial of degree n	$A_n t^n + A_{n-1}t^{n-1} + \cdots + A_0$	Include ALL terms down to constant
$Ke^{\alpha t}$	$Ae^{\alpha t}$	$A$ is unknown constant to find
$K_1\cos(\omega t) + K_2\sin(\omega t)$	$A\cos(\omega t) + B\sin(\omega t)$	Use BOTH cos and sin, even if g(t) has only one
Combination (e.g., $e^t \cos(t)$)	Use superposition: find $y_{p1}$ and $y_{p2}$ separately	Add solutions: $y_p = y_{p1} + y_{p2}$

The Golden Rule

      Your guess must have the right FORM. If g(t) is an exponential, your guess should be exponential. If g(t) is a polynomial, your guess should be polynomial.

When Your Guess Goes Wrong: Resonance

Critical Rule

      If your initial guess is already in $y_c$, it will NOT work! You MUST multiply by $t$ (or $t^2$ if still overlapping).

Modification Rules:

?

NORMAL (No overlap with $y_c$)

y_p = Ae^{\alpha t}

⚠

SINGLE RESONANCE (Guess is in $y_c$)

y_p = Ate^{\alpha t} \quad \text{(multiply by } t \text{)}

⚠⚠

DOUBLE RESONANCE (Both $e^{\alpha t}$ and $te^{\alpha t}$ in $y_c$)

y_p = At^2e^{\alpha t} \quad \text{(multiply by } t^2 \text{)}

Key Insight

      This is the SINGLE MOST IMPORTANT concept students miss! Always check whether your guess is in $y_c$ before substituting.

Example 1: Polynomial Forcing

Problem: Solve $y'' - 5y' + 6y = 2t$

STEP 1: Find $y_c$

r^2 - 5r + 6 = 0 \Rightarrow (r-2)(r-3) = 0 \Rightarrow r_1=2, r_2=3

y_c = C_1e^{2t} + C_2e^{3t}

STEP 2: Guess $y_p$ (g(t) is degree 1 polynomial)

y_p = At + B \quad \text{(include constant term!)}

Overlap check: $At + B$ is NOT in $y_c$ ✓

STEP 3: Substitute & solve

y_p' = A, \quad y_p'' = 0

0 - 5A + 6(At + B) = 2t

6At + (6B - 5A) = 2t

Matching: $6A = 2 \Rightarrow A = 1/3$, and $6B - 5A = 0 \Rightarrow B = 5/18$

y = C_1e^{2t} + C_2e^{3t} + \frac{1}{3}t + \frac{5}{18}

Example 2: Exponential (No Overlap)

Problem: Solve $y'' - 7y' + 12y = 4e^{2t}$

STEP 1: Find $y_c$

r^2 - 7r + 12 = 0 \Rightarrow r = 3, 4

y_c = C_1e^{3t} + C_2e^{4t}

STEP 2: Guess $y_p$ (g(t) is $e^{2t}$)

y_p = Ae^{2t}

Overlap check: $e^{2t}$ is NOT in $y_c$ (we have $e^{3t}$ and $e^{4t}$) ✓

STEP 3: Substitute & solve

y_p'' - 7y_p' + 12y_p = 4Ae^{2t} - 7(2A)e^{2t} + 12Ae^{2t} = 4e^{2t}

(4A - 14A + 12A)e^{2t} = 4e^{2t} \Rightarrow 2A = 4 \Rightarrow A = 2

y = C_1e^{3t} + C_2e^{4t} + 2e^{2t}

Example 3: Exponential with Resonance!

Problem: Solve $y'' - 7y' + 12y = 5e^{4t}$

STEP 1: Find $y_c$

y_c = C_1e^{3t} + C_2e^{4t}

STEP 2: Initial guess?

⚠ PROBLEM: If we try $y_p = Ae^{4t}$, this is already in $y_c$! RESONANCE!

y_p = Ate^{4t} \quad \text{(multiply by } t \text{)}

STEP 3: Substitute the modified guess

y_p = Ate^{4t} \Rightarrow y_p' = Ae^{4t} + 4Ate^{4t} \Rightarrow y_p'' = 8Ae^{4t} + 16Ate^{4t}

Substituting into ODE and simplifying (many terms cancel), we get: $2Ae^{4t} = 5e^{4t} \Rightarrow A = 5/2$

y = C_1e^{3t} + C_2e^{4t} + \frac{5}{2}te^{4t}

Physical Meaning

      The $te^{4t}$ term grows linearly with time! This represents resonance — the forcing frequency matches a natural frequency of the system.

Example 4: Double Resonance

Problem: Solve $y'' - 8y' + 16y = 2e^{4t}$

STEP 1: Find $y_c$

r^2 - 8r + 16 = 0 \Rightarrow (r-4)^2 = 0 \Rightarrow r = 4 \text{ (double)}

y_c = (C_1 + C_2t)e^{4t}

STEP 2: Initial guess?

⚠ PROBLEM: $e^{4t}$ AND $te^{4t}$ are BOTH in $y_c$! NEED $t^2$!

y_p = At^2e^{4t} \quad \text{(multiply by } t^2 \text{)}

STEP 3: Substitute & solve

After computing derivatives and substituting:

2Ae^{4t} = 2e^{4t} \Rightarrow A = 1

y = (C_1 + C_2t + t^2)e^{4t}

Example 5: Trigonometric Forcing

Problem: Solve $y'' - 2y' + y = 5\cos(2t) + 10\sin(2t)$

STEP 1: Find $y_c$

r^2 - 2r + 1 = 0 \Rightarrow (r-1)^2 = 0

y_c = (C_1 + C_2t)e^t

STEP 2: Guess $y_p$ (g(t) has sin and cos at frequency $\omega=2$)

y_p = A\cos(2t) + B\sin(2t)

Overlap check: This is NOT in $y_c$ (which involves $e^t$, not trig functions) ✓

STEP 3: Substitute & solve

$y_p' = -2A\sin(2t) + 2B\cos(2t)$
$y_p'' = -4A\cos(2t) - 4B\sin(2t)$

Substituting and collecting cos/sin terms gives system:

-3A - 4B = 5 \quad \text{and} \quad 4A - 3B = 10

Solving: $A = 1, B = -2$

y = (C_1 + C_2t)e^t + \cos(2t) - 2\sin(2t)

Example 6: Trigonometric Resonance!

Problem: Solve $y'' + 4y = 8\cos(2t)$

STEP 1: Find $y_c$

r^2 + 4 = 0 \Rightarrow r = \pm 2i

y_c = C_1\cos(2t) + C_2\sin(2t)

STEP 2: Initial guess?

⚠ PROBLEM: $\cos(2t)$ and $\sin(2t)$ are already in $y_c$! RESONANCE!

y_p = t[A\cos(2t) + B\sin(2t)] \quad \text{(multiply by } t \text{)}

STEP 3: Substitute & solve

After derivatives and substitution, the $y_p$ and $y_p''$ terms cancel (resonance magic!), leaving:

-4A\sin(2t) + 4B\cos(2t) = 8\cos(2t)

Thus: $A = 0, B = 2$

y = C_1\cos(2t) + C_2\sin(2t) + 2t\sin(2t)

Physical Resonance

      The amplitude of oscillation grows linearly with time ($2t$ multiplier)! This is true resonance — the driving force matches the natural oscillation frequency. Eventually the system will break or saturate!

Handling Multiple Terms: Superposition

Superposition Principle

      If $g(t) = g_1(t) + g_2(t)$, find $y_{p1}$ and $y_{p2}$ separately, then add them:
    
      y_p = y_{p1} + y_{p2}

Example: $y'' - 4y' = te^t + \cos(2t)$

STEP 1: Find $y_c$

r^2 - 4r = 0 \Rightarrow r(r-4) = 0 \Rightarrow r=0, 4

y_c = C_1 + C_2e^{4t}

STEP 2: Split into two problems

Problem 1: $y'' - 4y' = te^t$

Guess: $y_{p1} = (At + B)e^t$ (polynomial × exponential)

Problem 2: $y'' - 4y' = \cos(2t)$

Guess: $y_{p2} = A\cos(2t) + B\sin(2t)$ (trig functions)

STEP 3: Solve each separately, then combine

y_p = y_{p1} + y_{p2}

Decision Flowchart: What to Guess?

START: Look at g(t). What is its form?

Polynomial (degree n)

Guess: $A_n t^n + \cdots + A_0$

Exponential $e^{\alpha t}$

Guess: $Ae^{\alpha t}$

Trigonometric sin/cos

Guess: $A\cos + B\sin$

CHECK: Is your guess already in $y_c$?

NO

Use your guess as-is and substitute

YES

Multiply by $t$ (or $t^2$ if still overlapping)

SUBSTITUTE $y_p$ into ODE: $ay_p'' + by_p' + cy_p = g(t)$

MATCH COEFFICIENTS and solve for unknowns (A, B, etc.)

DONE! Your particular solution is $y_p$. General: $y = y_c + y_p$

Pitfalls to Avoid

Mistake 1: Incomplete Guesses for Polynomials

If $g(t) = 3t + 5$ (degree 1), your guess must be $At + B$, NOT just $At$!
If $g(t) = t^2 - 1$, guess $At^2 + Bt + C$ (all three terms needed)
Missing constants or terms will make your solution wrong.

Mistake 2: Forgetting the Resonance Check

Substituting $y_p = Ae^{\alpha t}$ when it's already in $y_c$ will give you 0 = g(t), which is impossible!
ALWAYS check: "Is my guess in $y_c$?"
If yes, multiply by $t$ immediately before substituting.

Mistake 3: Missing Both Sin and Cos

If $g(t) = 5\sin(3t)$ (only sine), you STILL need $y_p = A\cos(3t) + B\sin(3t)$ (both).
The cosine term will emerge from the derivatives of sine in the ODE!

Mistake 4: Solving the ODE First Without $y_c$

You MUST find $y_c$ first to check for resonance!
Skipping this leads to impossible guesses that don't work.

Key Takeaways: Quick Reference

Polynomial

g(t) = polynomial of degree n

Guess: $A_n t^n + A_{n-1}t^{n-1} + \cdots + A_0$

Include ALL terms!

Exponential

g(t) = $K e^{\alpha t}$

Guess: $Ae^{\alpha t}$

If $e^{\alpha t} \in y_c$: multiply by $t$

Trigonometric

g(t) = sin/cos at $\omega$

Guess: $A\cos(\omega t) + B\sin(\omega t)$

Use BOTH functions!

Combination

g(t) = sum of above types

Use SUPERPOSITION: solve each part, add results

Works because ODE is linear

When This Method FAILS

      This method only works for "nice" g(t): polynomials, exponentials, sines, cosines, and products. For g(t) like $\tan(t)$, $\ln(t)$, $1/t$, or other transcendental functions, use Variation of Parameters (next lecture!).

The Golden Rules
Find $y_c$ FIRST (homogeneous solution)
Guess $y_p$ with the right FORM based on g(t)
Check if guess is in $y_c$ → if yes, multiply by $t$
Substitute and match coefficients
General solution: $y = y_c + y_p$