Power series solutions — ordinary points — Hinglish notes

4.6.17 · Maths › Ordinary Differential Equations

Chapter: Ordinary Differential Equations Subtopic: Power series solutions at an ordinary point

METHOD KAISE CHALTA HAI (the recipe)

Worked Example 1 — sabse simple case

$y^{''} + y = 0$ ko $x_{0} = 0$ ke baare mein solve karo. (Hum jaante hain answer $cos x, sin x$ hai — Forecast-then-Verify ke liye perfect.)

Forecast: Mujhe expect hai ki $a_{0}$ -series $= cos x$ aur $a_{1}$ -series $= sin x$ hogi.

Maano $y = \sum a_{n} x^{n}$ , toh $y^{''} = \sum_{n \geq 2} n (n - 1) a_{n} x^{n - 2}$ .

Substitute karo: $\sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n - 2} + \sum_{n = 0}^{\infty} a_{n} x^{n} = 0.$

Pehli sum ko re-index karo $k = n - 2$ ke saath (toh $n = k + 2$ ): Kyun? — taaki dono sums $x^{k}$ carry karein. $\sum_{k = 0}^{\infty} (k + 2) (k + 1) a_{k + 2} x^{k} + \sum_{k = 0}^{\infty} a_{k} x^{k} = 0.$

$x^{k}$ ka coefficient zero hona chahiye: $(k + 2) (k + 1) a_{k + 2} + a_{k} = 0 \Rightarrow a_{k + 2} = - \frac{a _{k}}{( k + 2 ) ( k + 1 )} .$ Kyun? — ek single power series zero hoti hai tabhi jab har coefficient zero ho.

Ab calculate karo:

Even: $a_{2} = - \frac{a _{0}}{2 !}$ , $a_{4} = - \frac{a _{2}}{4 \cdot 3} = \frac{a _{0}}{4 !}$ , $a_{6} = - \frac{a _{0}}{6 !}$ …
Odd: $a_{3} = - \frac{a _{1}}{3 !}$ , $a_{5} = \frac{a _{1}}{5 !}$ …

Toh $y = a_{0} c o s x (1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots) + a_{1} s i n x (x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots) .$ Verified ✔ — forecast bilkul sahi nikla.

Worked Example 2 — Legendre-type, $(1 - x^{2}) y^{''} - 2 x y^{'} + 6 y = 0$ about $x_{0} = 0$

$x_{0} = 0$ legal kyun hai: upar dikhaya ja chuka hai ki ye ordinary point hai.

$y = \sum a_{n} x^{n}$ ko har piece mein plug karo. Polynomial coefficients ko andar multiply karo:

$y^{''} \sum n (n - 1) a_{n} x^{n - 2} - x^{2} y^{''} \sum n (n - 1) a_{n} x^{n} - 2 x y^{'} \sum 2 n a_{n} x^{n} + 6 y 6 \sum a_{n} x^{n} = 0.$

Pehli sum ko re-index karo ( $k = n - 2$ ); baaki already $x^{n}$ carry karte hain ( $n \to k$ rename karo):

$\sum_{k} [(k + 2) (k + 1) a_{k + 2} - k (k - 1) a_{k} - 2 k a_{k} + 6 a_{k}] x^{k} = 0.$

Aakhri teen ko combine kyun karein? Ye sab $a_{k} x^{k}$ multiply karte hain, toh: $- k (k - 1) - 2 k + 6 = - (k^{2} - k + 2 k - 6) = - (k^{2} + k - 6) = - (k + 3) (k - 2) .$

Recurrence: $(k + 2) (k + 1) a_{k + 2} = (k + 3) (k - 2) a_{k} \Rightarrow a_{k + 2} = \frac{( k + 3 ) ( k - 2 )}{( k + 2 ) ( k + 1 )} a_{k} .$

Note karo: $k = 2$ deta hai $a_{4} = 0$ , toh saare bade even terms vanish ho jaate hain — $a_{0}$ branch terminate ho jaata hai:

$a_{2} = \frac{( 3 ) ( - 2 )}{2 \cdot 1} a_{0} = - 3 a_{0}$ , $a_{4} = 0$ .
$y_{even} = a_{0} (1 - 3 x^{2})$ — ek polynomial solution (isliye Legendre equations se Legendre polynomials milte hain!).

Odd branch ( $a_{1}$ ) terminate nahi hoti (numerator $(k + 3) (k - 2)$ odd $k \geq 1$ ke liye kabhi $0$ nahi hota):

$a_{3} = \frac{( 4 ) ( - 1 )}{3 \cdot 2} a_{1} = - \frac{2}{3} a_{1}$ .
$a_{5} = \frac{( 5 ) ( 1 )}{5 \cdot 4} a_{3} = \frac{6}{20} a_{3} = \frac{3}{10} a_{3} = \frac{3}{10} (- \frac{2}{3}) a_{1} = - \frac{1}{5} a_{1}$ . Ye step kyun? — recurrence apply karo $k = 3$ ke saath: numerator $(k + 3) (k - 2) = 6 \cdot 1 = 6$ , denominator $(k + 2) (k + 1) = 5 \cdot 4 = 20$ , toh factor $= 6/20 = 3/10$ ; phir $a_{3} = - \frac{2}{3} a_{1}$ substitute karo.

$y = a_{0} (1 - 3 x^{2}) + a_{1} (x - \frac{2}{3} x^{3} - \frac{1}{5} x^{5} - \dots) .$

Figure — Power series solutions — ordinary points

Common mistakes (Steel-man + fix)

Active recall

Recall Self-test (dekhne se pehle try karo)

Ordinary point define karo.
$a_{0}, a_{1}$ free kyun hote hain?
Fuchs' theorem radius of convergence ke baare mein kya guarantee karta hai?
Example 2 mein even series terminate kyun hua?

Answers: 1. Standard form mein $P, Q$ , $x_{0}$ par analytic hain. 2. Recurrence sirf $a_{k}$ se $a_{k + 2}$ deta hai; pehle do unconstrained = do integration constants. 3. $\geq$ distance from $x_{0}$ to nearest singular point. 4. Recurrence ka numerator $(k + 3) (k - 2)$ , $k = 2$ par $0$ ho gaya, $a_{4}$ aur saare bade even terms ko zero kar diya.

Recall Feynman: 12-saal ke bacche ko samjhao

Kuch equations ek curve ko uski steepness aur curviness se connect karti hain. Hum unka answer normal functions se nahi likh sakte. Toh hum guess karte hain ki answer ek giant polynomial hai — $a_{0} + a_{1} x + a_{2} x^{2} + \dots$ — lekin numbers nahi pata. Jab hum ye guess equation mein daalte hain, toh equation simple sums ki ek chain ban jaati hai jo kehti hai: "agar tum mujhe pehle do numbers batao, main baaki sab calculate kar dunga." Toh hum pehle do freely choose karte hain (ye hamare do free choices hain), aur machine poori infinite list nikaal deti hai. Jab tak equation hamare starting point ke paas "broken" nahi hai (dividing-by-zero spot nahi hai), ye guessed polynomial actually kaam karta hai.

Flashcards

y^{''} + P y^{'} + Q y = 0

mein

x_{0}

ko ordinary point kya banata hai?

Dono

P

aur

Q

x_{0}

par analytic hain (convergent power series hai).

Exactly do free constants

a_{0}, a_{1}

kyun aate hain?

Recurrence sirf

a_{k}

a_{k + 2}

determine karta hai,

a_{0}, a_{1}

undetermined rehte hain = 2nd-order ODE ke do integration constants.

Fuchs' theorem ordinary point par kya guarantee karta hai?

Do linearly independent power-series solutions exist karte hain, jo kam se kam nearest singular point tak convergent hain.

Point classify karne se pehle pehla step kya hai?

ODE ko standard form

y^{''} + P y^{'} + Q y = 0

mein daalo (leading coefficient se divide karo).

y^{''} + y = 0

ke liye recurrence kya hai?

a_{k + 2} = - \frac{a _{k}}{( k + 2 ) ( k + 1 )}

, jo

cos x

aur

sin x

deta hai.

Substitute karne ke baad sums ko re-index kyun karte hain?

Taaki har sum same power

x^{k}

carry kare, coefficients combine ho sakein aur zero set kiye ja sakein.

Power-series solution polynomial mein terminate kab hoti hai?

Jab recurrence ka numerator kisi

k

par zero ho jaata hai, us branch ke saare aage ke coefficients zero ho jaate hain (jaise Legendre polynomials).

Radius of convergence ka relation kya hai?

R \geq

distance from

x_{0}

to nearest singular point in the complex plane.

Connections

Singular points and Frobenius method — jab $P, Q$ analytic nahi hote tab kya karein.
Legendre's equation and Legendre polynomials — terminating power series.
Taylor series and analyticity — "analytic" ka matlab.
Existence and uniqueness for linear ODEs — Fuchs' theorem ka context.
Recurrence relations — yahan algebraic engine.
Hermite and Airy equations — classic ordinary-point examples.

Power series solutions — ordinary points

The big picture

Ordinary point kya hota hai?