Second-Order Linear ODEs – Superposition, Linear Independence & Characteristic Roots

Two Dependent Variables & Uncertainty

• Opening remark: when a system contains two dependent variables, each can influence the outcome; uncertainty arises because what happens to one need not happen to the other.
• Practical implication: always expect multiple possible behaviors unless a theorem guarantees uniqueness.

Fast Solution of Differential Equations

• Goal: “find a quick way of solving” – the characteristic-equation method for linear, homogeneous, constant-coefficient ODEs.
• Key observation: focus on one function that is unique (the lecturer jokingly calls it “the delivery when antidelivery is safe”).

Superposition Principle (c₁, c₂‐Form)

• For a second-order homogeneous linear ODE $a\,y''+b\,y'+c\,y=0$ every linear combination of two independent solutions is again a solution.
• Statement: “Pick any $c<em>1, c</em>2$, the sum $c<em>1y</em>1+c<em>2y</em>2$ is still a solution; no restriction on $c<em>1,c</em>2$.”
  → wording change: “for some $c<em>1,c</em>2$” vs “for all $c<em>1,c</em>2$” distinguishes existence from universality.
• Humorous example: “Pick your name for $\cos x$, your friend’s name for $\sin x$—any coefficients will work.”

“Two-Fold Infinity” of Solutions

• Idea: allowing all real values of $c<em>1,c</em>2$ yields an uncountable ("two-fold infinity") family of solutions.
  – Visualised as ${c<em>1,c</em>2}\cong \mathbb R^2$.

English vs Mathematical “Independence”

• Everyday: independent ⇒ unique, incomparable.
• Mathematical (linear) independence: a set ${f<em>1,f</em>2}$ on interval $I$ is linearly independent if neither is a constant multiple of the other on $I$.
  – Example independent pair: $\sin x,\;\cos x$ (quotient gives $\tan x$ or $\cot x$, not a constant).
  – Example dependent pair: $x,\;2x$ (constant multiple).

Quick Tests for Independence

• Constant-multiple test: if $f(x)=k\,g(x)$ for some constant $k$ on $I$ ⇒ dependent.
• Wronskian (to be formalised later): $W(f,g)(x)=f g' - f' g$.
  – If $W\not\equiv 0$ on $I$ ⇒ independent.

Theorem 4 (Existence Version)

• If $y$ is any solution on interval $I$, there exists constants $c<em>1,c</em>2$ such that $y=c<em>1y</em>1+c<em>2y</em>2$ where $y<em>1,y</em>2$ are a fundamental set on $I$.
• Not universal for all $c<em>1,c</em>2$ – merely one pair fits each specific solution.

Domain & Piecewise Issues

• Example: solutions $y<em>1=x,\;y</em>2=x^3$ may each satisfy initial data on overlapping pieces, yet the whole domain may require piecewise definition.
• Strategy: compute separate Wronskians $W<em>L, W</em>R$ on left/right sub-intervals, then glue solutions.

Characteristic-Equation Method (Section 3.1 “new stuff”)

• Start with ODE $a\,y''+b\,y'+c\,y=0$.
• Form characteristic polynomial $a r^2 + b r + c = 0$; roots $r<em>1,r</em>2$ dictate the template of the general solution.

Case 1 – Distinct, Real Roots ($r<em>1\ne r</em>2\in\mathbb R$)

• Theorem 5: $y=c<em>1 e^{r</em>1 x}+c<em>2 e^{r</em>2 x}$.

Case 2 – Repeated, Real Root ($r<em>1=r</em>2=r$)

• Solution: $y=(c<em>1+c</em>2 x)\,e^{r x}$.

Case 3 – Complex Conjugate Roots ($r=\alpha\pm i\beta,\;\beta\ne 0$)

• Write $e^{(\alpha+i\beta)x}=e^{\alpha x}(\cos \beta x + i\sin \beta x)$.
• Real general solution: $y=e^{\alpha x}\left(c<em>1\cos \beta x + c</em>2\sin \beta x\right)$.

Extensions to Higher Order

• Same projection idea: reduce an $n^{\text{th}}$-order constant-coefficient ODE to an $n$-degree polynomial; solve roots; build exponentials/polynomials/sines-cosines accordingly.
• Lecturer hints at “projection to 2-D philosophy” – treat each pair of complex roots as a 2-D subsystem.

Theorems Beyond Second Order

• Two final theorems (names not provided) cover:

1. Multiple distinct real roots in high order.
2. Complex roots in conjugate pairs.
3. Repeated roots requiring extra $x$ factors (up to multiplicity $k$ gives $x^{k-1}e^{rx}$ terms).

Practical Workflow for Students

1. Verify whether the ODE is linear, homogeneous, constant-coefficient.
2. Write the characteristic polynomial, factor (use technology if degree >2).
3. Classify roots (distinct, repeated, complex).
4. Construct template solution as per the three cases above.
5. Impose initial conditions to solve for $c<em>1,c</em>2$.
6. If domain is piecewise, compute solution/Wronskian on each sub-interval.

Examples & Metaphors Mentioned

• “Your name $\cos x$, friend’s name $\sin x$” illustrates arbitrary constants.
• Dividing $\sin x$ by $\cos x$ gives $\tan x$ – shows non-multiplicity yet relationship.
• “Two-fold infinity” describes the $\mathbb R^2$ space of coefficients.

Ethical / Pedagogical Comments

• Lecturer avoids giving problems that demand heavy symbolic integration late in the course – focuses on conceptual mastery.
• Emphasis on not “wasting time to re-prove long-established theorems” in class; instead supply convincing examples.

Key Vocabulary

• Dependent Variable – quantity whose value depends on others.
• Superposition – linear combination of solutions is a solution.
• Fundamental Set – minimal independent set spanning solution space.
• Linear Independence – no constant multiple relation; Wronskian non-zero.
• Characteristic Equation – algebraic equation $ar^2+br+c=0$ for 2nd order.
• Repeated Root – root multiplicity >1, introduces polynomial factors.
• Complex Conjugate Roots – pair producing oscillatory $\sin,\cos$ terms.

Numerical / Symbolic Facts to Memorise

• $W(f,g)(x)=f g' - f' g$ (Wronskian for two functions).
• Template summary:
  – Distinct real: $c<em>1e^{r</em>1x}+c<em>2e^{r</em>2x}$.
  – Repeated real: $(c<em>1+c</em>2x)e^{rx}$.
  – Complex pair $\alpha\pm i\beta$: $e^{\alpha x}(c<em>1\cos\beta x+c</em>2\sin\beta x)$.

Closing Advice

• Always test for independence early; without it, superposition cannot span entire solution space.
• Use technology for root-finding but understand the template shapes.
• Expect exam questions on:
  – Identifying independence via Wronskian.
  – Writing general solutions for the three root patterns.
  – Applying initial/boundary conditions.
  – Recognising when piecewise treatment is needed (domain breaks).