Variation of Parameters, Power-Series Methods & Analytic Points

Variation of Parameters & Product Rule

• Instructor begins by manipulating terms such as $2ax\cdot x$ and $2ax^2\cdot x$ to illustrate the product rule and set up the method of variation of parameters.
• Variation of parameters is introduced as a
  • Theorem from Section 3.5 (called the “variation of perimeter/parameters theorem”).
  • Preferred over earlier "trial-and-error" methods for finding particular solutions of non-homogeneous linear ODEs.
• Key reminder: a product rule application will often generate extra terms and may look tedious at first, but it systematically yields the correct integrals for the parameter functions.

Methods of Solving Differential Equations Reviewed

• Three approaches mentioned so far:
  1. "Trial and error" (undetermined coefficients or guessing) — not a systematic method.
  2. Method of annihilators / method of undetermined coefficients.
  3. Variation of parameters — main focus today.
• Goal: reduce "tedious" algebra by choosing the method that fits the structure of the ODE.

Wronskian and Linear Independence

• For a second-order linear ODE with fundamental solution $y<em>1 = \cos x$, the second independent solution $y</em>2$ is usually found by reduction of order.
• The Wronskian for two solutions is $W(y<em>1,y</em>2)=\begin{vmatrix}y<em>1 & y</em>2\ y<em>1' & y</em>2'\end{vmatrix}$.
• When the ODE is of order 3, one must build a $3\times3$ Wronskian:
  $W(y<em>1,y</em>2,y<em>3)=\begin{vmatrix}y</em>1 &amp; y<em>2 & y</em>3\ y<em>1' & y</em>2' &amp; y<em>3'\ y</em>1'' &amp; y<em>2'' & y</em>3''\end{vmatrix}.$
  The instructor notes this is beyond the required scope for the next test and will be postponed.

Syllabus Updates / Sections NOT on the Next Test

• Sections 3.4, 3.6, 3.7, 3.8 are officially removed from the second test.
• Nevertheless, Section 3.8 (end-point & eigenvalue problems) contains interesting physics applications; students are encouraged to read for enrichment.

Smoothness, Continuity & Differentiability

• "Not-smooth" functions: continuous but not differentiable at one or more points.
• Vocabulary:
  • Base continuous ≡ continuous everywhere on the interval.
  • Corner point ≡ point where the derivative fails to exist (e.g.
    $|x|$ at $x=0$).
• Two standard conditions that may fail:
  1. The function value exists but derivative does not.
  2. The derivative exists but is discontinuous at a point.
• Reminder: to prove differentiability you must check an open interval; closed-endpoints can hide vertical tangents ($\pm\infty$ slopes).

Endpoint Problems & Eigenvalues (Guitar-String Model)

• Typical boundary-value problem (BVP): $y'' + \lambda y = 0$ with $y(0)=0$ and $y(\pi)=0$.
  • Physical model: a guitar string fixed at $x=0$ and $x=\pi$.
  • Leads to quantised eigenvalues $\lambda_n = n^2$ and eigenfunctions $\sin(nx)$.
• Students asked to study Example 1 (p. 216) and Example 2 (p. 217) in the text to see how solutions change when boundary conditions are applied in the interior versus at end-points.
• Example 3 (p. 224) involves hyperbolic geometry ($\sinh,\cosh$). Reading is optional if you have not yet studied hyperbolic functions.

Reading Recommendations

• Read p. 216 – 224 for a “bigger picture” of eigenvalues and end-point problems.
• Physics majors will see direct applications (e.g.
  normal modes, quantum wells).

Linear-Algebra Primer to Come

• A one-session crash course on matrices is planned before returning to Chapter 5.
  • Motivation: many DE topics (Wronskians, systems, variation of parameters, series solutions) rely on matrix language.

Power-Series Methods (Chapter 8)

Why Power Series?

• Sometimes a separable or exact method is faster than a series; however, certain ODEs have no elementary closed-form and require a series solution.
• The central idea: assume $y=\sum<em>{n=0}^\infty a</em>n x^n$, substitute into the ODE, match coefficients, and solve for $a_n$ recursively.

Term-by-Term Differentiation & Integration

• A power series $\sum a_n x^n$ represents an analytic function on its interval of convergence.
• Because differentiation is distributive over addition and scalar multiples, we may differentiate “term-wise”:
  $\bigl(\sum<em>{n=0}^\infty a</em>n x^n\bigr)' = \sum<em>{n=1}^\infty n a</em>n x^{n-1}.$
• Likewise for indefinite integration.

Radius of Convergence

• Determined by the ratio test:
  $L = \lim<em>{n\to\infty}\left|\frac{a</em>{n+1}}{a_n}\right|,$
  $R = \frac{1}{L}.$
• Convergence properties:
  • If |x|<R, the series converges absolutely.
  • If |x|>R, it diverges.
  • If $|x|=R$, separate testing is required.
• The ratio test is preferred over the root test in this chapter.

Analytic, Ordinary & Singular Points — Correct Logical Order

1. Analytic function: expandable in a convergent power series in some open interval around $x_0$.
2. Ordinary point of an ODE: the coefficients of the highest derivative are analytic and non-zero at $x_0$.
3. Singular point: any point that is not ordinary.
   • Many textbooks present the order (3 → 2 → 1); instructor advises mentally reversing it (1 → 2 → 3) for clarity.

Index Shifting Example

• Typical manipulation: convert (a{n-1}x^{n-1}) to a common (x^n) term by $\sum</em>{n=1}^\infty a<em>{n-1}x^{n-1} = \sum</em>{n=0}^\infty a_n x^n\quad (\text{via }n\to n+1).$
• Index shifts must be justified, usually by writing out the first few terms and showing equality.

Caveats

• Plugging $n=0,1,2,\dots$ into the recursion may not immediately reveal a closed-form for $a_n$; patience and pattern recognition are essential.
• A seemingly simple ODE might yield an odd-power series only, or even yield two interleaved series (even/odd).

Function Parity: Odd vs Even

• Odd function: $f(-x)=-f(x)$ (e.g. $\sin x$).
• Even function: $f(-x)=f(x)$ (e.g. $\cos x$).
• Some texts phrase “odd” as “not even,” but it is more precise to use the algebraic definition.

Number-Theory Tangent: Primes, Composites & Larger Infinities

• A composite number is simply one that is not prime.
• The classic proof that prime numbers are infinite (Euclid’s proof) is recommended reading.
• In set theory, different “sizes” of infinity exist (e.g. $\aleph_0$ vs. continuum). Instructor hints at Cantor’s results (“bigger than infinity”).

Continuity, Differentiability & Vertical Tangents

• For a function to be differentiable on $[a,b]$ it must actually be differentiable on the open interval $(a,b).$ Endpoints often cause vertical tangents or undefined derivatives.
• Visual test: if left and right limits of the derivative disagree or are infinite, the point is non-differentiable.

Practical Advice & Study Tips

• Read 8.1 thoroughly; master the radius-of-convergence machinery.
• Work through one “tedious” full power-series solution yourself to internalise index-shifting and coefficient matching.
• Keep linear-algebra notes handy (matrix operations, determinants, eigenvalues) — they recur throughout Chapters 5 & 8.
• When choosing a method:
  • If an ODE is separable or exact, do that before trying a series.
  • Use series when coefficients are non-polynomial or possess singular points.
• Physics students: link eigenvalue BVPs to standing waves, quantum mechanics, and heat/vibration problems.

(End of compiled lecture notes.)