27-03 — Second-Order Linear Homogeneous ODEs

Phase: Ordinary Differential Equations | Subject: 27-03 Prerequisites: 27-02-first-order-odes.md, 03-06-complex-numbers.md Next subject: 27-04-second-order-nonhomogeneous-odes.md

Learning Objectives

By the end of this subject, you will be able to:

1. Write the general form of a 2nd-order linear homogeneous ODE: $ay'' + by' + cy = 0$
2. Solve using the characteristic equation $ar^2 + br + c = 0$
3. Handle three cases: distinct real roots, repeated roots, complex conjugate roots
4. Interpret solutions physically: damped harmonic oscillator
5. Apply initial conditions to find particular solutions

Core Content

General Form and Superposition

$$ay'' + by' + cy = 0$$

where $a \neq 0$, $a, b, c$ are constants (constant-coefficient case).

Principle of Superposition: If $y_1(x)$ and $y_2(x)$ are solutions, then any linear combination $c_1 y_1 + c_2 y_2$ is also a solution.

The Characteristic Equation

Assume solutions of the form $y = e^{rx}$. Substituting:

$ar^2 e^{rx} + br e^{rx} + c e^{rx} = 0 \Rightarrow e^{rx}(ar^2 + br + c) = 0$.

Since $e^{rx} \neq 0$, we need:

$$ar^2 + br + c = 0$$

This is the characteristic equation.

Three Cases

Case 1: Distinct Real Roots ($b^2 - 4ac > 0$)

Roots: $r_1 \neq r_2$, both real. General solution: $y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$.

Case 2: Repeated Real Root ($b^2 - 4ac = 0$)

Root: $r = -b/(2a)$ (double). General solution: $y(x) = (c_1 + c_2 x) e^{rx}$.

Case 3: Complex Conjugate Roots ($b^2 - 4ac < 0$)

Roots: $r = \alpha \pm i\beta$ where $\alpha = -b/(2a)$, $\beta = \sqrt{4ac - b^2}/(2a)$. General solution: $y(x) = e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$.

Physical Interpretation: Damped Harmonic Oscillator

$$m x'' + \gamma x' + kx = 0$$

Characteristic: $mr^2 + \gamma r + k = 0$.

• Overdamped ($\gamma^2 > 4mk$): $r_1, r_2 < 0$ real, monotonic decay
• Critically damped ($\gamma^2 = 4mk$): repeated negative root $r$, fastest decay without oscillation
• Underdamped ($\gamma^2 < 4mk$): complex roots with negative real part, decaying oscillation

Key Terms

• Characteristic Equation
• Superposition Principle
• Overdamped, Critically Damped, Underdamped
• Fundamental Solution Set

Worked Examples

Example 1: Distinct Real Roots

Solve $y'' - 5y' + 6y = 0$, $y(0) = 1$, $y'(0) = 0$.

Solution: Characteristic: $r^2 - 5r + 6 = 0 \Rightarrow (r-2)(r-3) = 0$. Roots: $r = 2, 3$. General: $y = c_1 e^{2x} + c_2 e^{3x}$. $y(0) = c_1 + c_2 = 1$. $y' = 2c_1 e^{2x} + 3c_2 e^{3x}$, $y'(0) = 2c_1 + 3c_2 = 0$. Solve: $c_1 = 3, c_2 = -2$. $y = 3e^{2x} - 2e^{3x}$.

Example 2: Repeated Root

Solve $y'' + 6y' + 9y = 0$.

Solution: Characteristic: $r^2 + 6r + 9 = 0 \Rightarrow (r+3)^2 = 0$. Repeated root $r = -3$. General: $y = (c_1 + c_2 x)e^{-3x}$.

Example 3: Complex Roots

Solve $y'' + 4y' + 13y = 0$.

Solution: Characteristic: $r^2 + 4r + 13 = 0$. Discriminant: $16 - 52 = -36$. $r = \frac{-4 \pm \sqrt{-36}}{2} = -2 \pm 3i$. General: $y = e^{-2x}(c_1 \cos 3x + c_2 \sin 3x)$.

Quiz

Q1: What does the concept of Overdamped primarily refer to in this subject?

A) The definition and application of Overdamped B) A historical anecdote about Overdamped C) A visual representation of Overdamped D) A computational error related to Overdamped

Correct: A)

• If you chose A: Overdamped is defined as: the definition and application of overdamped. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Overdamped is defined as: the definition and application of overdamped. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Overdamped is defined as: the definition and application of overdamped. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Overdamped is defined as: the definition and application of overdamped. The other options describe different aspects that are not the primary focus.

Q2: Which of the following is the key formula discussed in this subject?

A) ay'' + by' + cy = 0 B) A simplified version of ay'' + by' + cy = 0... C) An unrelated formula from a different topic D) The inverse operation of the formula in question

Correct: A)

• If you chose A: The formula ay'' + by' + cy = 0 is central to this subject. The other options are either simplified versions or unrelated. Correct!
• If you chose B: This is incorrect. The formula ay'' + by' + cy = 0 is central to this subject. The other options are either simplified versions or unrelated.
• If you chose C: This is incorrect. The formula ay'' + by' + cy = 0 is central to this subject. The other options are either simplified versions or unrelated.
• If you chose D: This is incorrect. The formula ay'' + by' + cy = 0 is central to this subject. The other options are either simplified versions or unrelated.

Q3: What is the primary purpose of Critically damped?

A) It is primarily a historical notation system B) It is used only in advanced research contexts C) It is used to critically damped in mathematical analysis D) It replaces all other methods in this domain

Correct: C)

• If you chose A: This is incorrect. Critically damped serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Critically damped serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Critically damped serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Critically damped serves the purpose described in the correct answer. The other options misrepresent its role.

Q4: Which statement about Underdamped is TRUE?

A) Underdamped is mentioned only as a historical footnote B) Underdamped is a fundamental concept covered in this subject C) Underdamped is not related to this subject D) Underdamped is an advanced topic beyond this subject's scope

Correct: B)

• If you chose A: This is incorrect. Underdamped is a fundamental concept covered in this subject. This subject covers Underdamped as part of its core content.
• If you chose B: Underdamped is a fundamental concept covered in this subject. This subject covers Underdamped as part of its core content. Correct!
• If you chose C: This is incorrect. Underdamped is a fundamental concept covered in this subject. This subject covers Underdamped as part of its core content.
• If you chose D: This is incorrect. Underdamped is a fundamental concept covered in this subject. This subject covers Underdamped as part of its core content.

Q5: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) The inverse of the correct answer C) A different result from a common mistake D) Repeated Root

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is Repeated Root. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is Repeated Root. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is Repeated Root. The other options represent common errors.
• If you chose D: The worked examples show that the result is Repeated Root. The other options represent common errors. Correct!

Q6: How are Underdamped and Characteristic Equation related?

A) Underdamped is the inverse of Characteristic Equation B) Underdamped and Characteristic Equation are completely unrelated topics C) Underdamped is a special case of Characteristic Equation D) Underdamped and Characteristic Equation are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Underdamped and Characteristic Equation are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Underdamped and Characteristic Equation are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Underdamped and Characteristic Equation are covered in this subject as interconnected topics.
• If you chose D: Both Underdamped and Characteristic Equation are covered in this subject as interconnected topics. Correct!

Q7: What is a common pitfall when working with Superposition Principle?

A) The main error with Superposition Principle is using it when it is not needed B) A common mistake is confusing Superposition Principle with a similar concept C) Superposition Principle has no common misconceptions D) Superposition Principle is always computed the same way in all contexts

Correct: B)

• If you chose A: This is incorrect. Students often confuse Superposition Principle with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Superposition Principle with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Superposition Principle with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Superposition Principle with similar-sounding or related concepts. Pay attention to the precise definitions.

Q8: When should you apply Fundamental Solution Set?

A) Avoid Fundamental Solution Set unless explicitly instructed B) Fundamental Solution Set is not practically useful C) Apply Fundamental Solution Set to solve problems in this subject's domain D) Use Fundamental Solution Set only in pure mathematics contexts

Correct: C)

• If you chose A: This is incorrect. Fundamental Solution Set is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Fundamental Solution Set is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Fundamental Solution Set is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Fundamental Solution Set is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Solve $y'' - y = 0$.
   

   Click for answer

   Characteristic: $r^2 - 1 = 0$, roots $r = \pm 1$. $y = c_1 e^x + c_2 e^{-x}$.

2. Solve $y'' + 4y = 0$.
   

   Click for answer

   Characteristic: $r^2 + 4 = 0$, roots $r = \pm 2i$. $y = c_1 \cos 2x + c_2 \sin 2x$.

3. Solve $4y'' - 4y' + y = 0$.
   

   Click for answer

   $4r^2 - 4r + 1 = 0 \Rightarrow (2r-1)^2 = 0$, $r = 1/2$ repeated. $y = (c_1 + c_2 x)e^{x/2}$.

4. Solve $y'' - 2y' + 5y = 0$, $y(0) = 1$, $y'(0) = -1$.
   

   Click for answer

   $r^2 - 2r + 5 = 0$, $r = 1 \pm 2i$. $y = e^x(c_1 \cos 2x + c_2 \sin 2x)$. $y(0) = c_1 = 1$. $y' = e^x[(c_1 + 2c_2)\cos 2x + (c_2 - 2c_1)\sin 2x]$. $y'(0) = c_1 + 2c_2 = 1 + 2c_2 = -1 \Rightarrow c_2 = -1$. $y = e^x(\cos 2x - \sin 2x)$.

5. The equation $y'' + 0.1y' + y = 0$ represents what type of damping?
   

   Click for answer

   $\gamma^2 = 0.01$, $4mk = 4(1)(1) = 4$. Since $0.01 < 4$, it's underdamped — decaying oscillation.

Summary

Key takeaways:

• Assume $y = e^{rx}$, form characteristic equation $ar^2 + br + c = 0$
• Distinct real roots: $c_1 e^{r_1 x} + c_2 e^{r_2 x}$
• Repeated root $r$: $(c_1 + c_2 x)e^{rx}$
• Complex roots $\alpha \pm i\beta$: $e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$
• Two initial conditions uniquely determine $c_1, c_2$
• Damped harmonic oscillator: overdamped, critically damped, or underdamped

Pitfalls

• Forming the wrong characteristic equation: For $ay'' + by' + cy = 0$, the characteristic equation is $ar^2 + br + c = 0$, not $ar^2 + br + c = y$ or $r^2 + br + c = 0$. Students sometimes forget the coefficient $a$ on $r^2$ or incorrectly write $r^2 + (b/a)r + (c/a) = 0$ but then solve for the wrong roots. Also, the term $y$ maps to $c \cdot e^{rx}$, so the constant $c$ stays as $c$ in the characteristic equation, not disappearing.
• Giving real-valued solutions for complex roots that don't span all solutions: When the characteristic equation yields complex conjugate roots $\alpha \pm i\beta$, the general solution is $e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$. A common error is writing only $e^{\alpha x}(c_1 \cos \beta x)$ (forgetting the sine term) or writing $c_1 e^{\alpha x}\cos\beta x + c_2 e^{\alpha x}\cos\beta x$ (two copies of cosine). Both $\cos$ and $\sin$ are needed to span the 2-dimensional solution space.
• Forgetting the $x$ factor for repeated roots: When the characteristic equation has a double root $r$, one solution is $e^{rx}$ and the second is $xe^{rx}$. A frequent mistake is writing $y = (c_1 + c_2)e^{rx}$, which collapses to a single constant $Ce^{rx}$ — this is only a 1-parameter family, not the general solution. The $c_2 x e^{rx}$ term is essential.
• Misidentifying damping type based on the discriminant of the wrong quadratic: For $my'' + \gamma y' + ky = 0$, the discriminant is $\gamma^2 - 4mk$, NOT $\gamma^2 - 4k$ or $b^2 - 4ac$ using coefficients from a different convention. Double-check which coefficients correspond to mass, damping, and stiffness before classifying as overdamped, critically damped, or underdamped.
• Solving for constants $c_1, c_2$ without using both initial conditions: A 2nd-order IVP requires two conditions, typically $y(0)$ and $y'(0)$. A common error is using $y(0)$ to find $c_1$, then forgetting to apply $y'(0)$ to find $c_2$, leaving the answer in terms of an unknown constant. Always use both conditions — and differentiate correctly before plugging in $x=0$.

Next Steps

Next up: 27-04-second-order-nonhomogeneous-odes.md — method of undetermined coefficients and variation of parameters.