Power series solutions — ordinary points: Visual walkthrough

4.6.17 · D2 · Maths › Ordinary Differential Equations › Power series solutions — ordinary points

Hum yeh seedha-saada equation solve karte hain $y^{''} + y = 0$ point $x_{0} = 0$ ke aas-paas, aur hum dekhte hain ki har symbol apni jagah kaise earn karta hai. Full recipe aur mistake list ke liye parent note Power Series Solutions — Ordinary Points dekho.

Step 1 — Power series kya hoti hai, aur isse guess kyun karte hain?

KYA HAI. Ek power series bas ek infinite lamba polynomial hota hai: $y (x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots = \sum_{n = 0}^{\infty} a_{n} x^{n} .$

Har symbol ko uski jagah pe padho:

$x$ — input, ek slider jise tum $0$ se left/right move kar sakte ho.
$a_{n}$ — ek fixed number (abhi unknown), term $x^{n}$ ka "weight."
$a_{n} x^{n}$ — ek bent curve; zyada $n$ matlab zyada bend, aur yeh tab hi active hota hai jab $x$ thoda bada ho.
$\sum_{n = 0}^{\infty}$ — "in saari bent curves ko add karo," $n = 0$ se shuru karke.

GUESS KyUN KARTE HAIN. $x = 0$ ke paas, powers size order mein pile up hoti hain: $1$ dominate karta hai, phir $x$ , phir $x^{2}$ , ... Toh pehle kuch numbers $a_{0}, a_{1}, a_{2}$ origin ke paas ka shape decide karte hain. Agar hum woh numbers find kar lein, toh curve pata chal jaata hai. Ek smooth curve = inhi simple pieces ka stack.

PICTURE. Dekho ki ek aur term add karne se curve kaise ek real function ke kareeb aata jaata hai.

Figure — Power series solutions — ordinary points

Step 2 — Differentiation us stack ke saath kya karta hai

KYA HAI. Equation $y^{''} + y = 0$ use karne ke liye humein $y^{''}$ chahiye, second derivative (slope khud kaise change hota hai — "curviness"). Ek term ko differentiate karna power rule use karta hai: $\frac{d}{d x} a_{n} x^{n} = n a_{n} x^{n - 1} .$

Term by term:

$n$ (front factor) — exponent se neeche aata hai aur coefficient se multiply ho jaata hai.
$x^{n - 1}$ — exponent ek se kam ho jaata hai, toh poori series ek power neeche shift ho jaati hai.

Yeh do baar karo: $y^{'} = \sum_{n = 1}^{\infty} n a_{n} x^{n - 1}, y^{''} = \sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n - 2} .$

START INDEX KyUN MOVE HOTA HAI. $y^{'}$ mein $n = 0$ term constant $a_{0}$ tha; uska derivative $0$ hai, toh woh drop out ho jaata hai — sum ab $n = 1$ se start hota hai. $y^{''}$ mein $n = 0$ aur $n = 1$ dono terms khatam ho jaati hain, toh yeh $n = 2$ se start hota hai. Lower limit decoration nahi hai; ise bhoolna tumhare pehle coefficients corrupt kar deta hai (parent note ki re-indexing mistake dekho).

PICTURE. Har derivative coefficient labels ko ek slot left slide karta hai aur unhe scale karta hai.

Step 3 — Substitute karo: ODE ko ek sum-equals-zero mein badlo

KYA HAI. Equation kehti hai $y^{''} + y = 0$ . Apni do series daalo: $yeh hai y^{''} n = 2 \sum \infty n (n - 1) a_{n} x^{n - 2} + yeh hai y n = 0 \sum \infty a_{n} x^{n} = 0.$

Left sum — powers $x^{0}, x^{1}, x^{2}, \dots$ carry karta hai lekin $x^{n - 2}$ likha hua hai.
Right sum — same powers carry karta hai lekin $x^{n}$ likha hua hai.
$= 0$ — yeh slider par $x$ ki har value ke liye hold karna chahiye, sirf ek ke liye nahi.

KyUN PROGRESS HAI. Ek differential equation (slopes aur curviness ke baare mein) ab ek statement ban gayi hai ki ek single infinite sum zero hai. Ab yeh ek algebra problem hai.

PICTURE. Do sums do rows of boxes hain; woh same powers carry karte hain lekin mismatched labels $n$ ke saath.

Step 4 — Re-index karo taaki dono rows ek hi power count karein

KYA HAI. Pehle sum mein power $x^{n - 2}$ hai; dummy counter ko $k = n - 2$ se rename karo, yaani $n = k + 2$ . Jahan bhi $n$ aaye, replace karo: $n (n - 1) a_{n} x^{n - 2} ⟶ (k + 2) (k + 1) a_{k + 2} x^{k} .$

$n \to k + 2$ factor mein: $n (n - 1) \to (k + 2) (k + 1)$ .
$a_{n} \to a_{k + 2}$ : coefficient index saath aata hai.
$x^{n - 2} \to x^{k}$ : ab clearly $x^{k}$ carry karta hai.
Start move hoti hai: $n = 2$ ban jaata hai $k = 0$ .

Doosra sum already $x^{n}$ carry karta hai; bas $n \to k$ rename karo. Dono ab $\sum_{k = 0}^{\infty} (\dots) x^{k}$ padhte hain: $\sum_{k = 0}^{\infty} [(k + 2) (k + 1) a_{k + 2} + a_{k}] x^{k} = 0.$

KyUN. Do series ko coefficient-by-coefficient tabhi add kar sakte ho jab woh same powers list karein. Re-indexing woh alignment hai jo ise legal banata hai.

PICTURE. Re-labelled boxes ab matching columns mein baithe hain, add hone ke liye ready.

Step 5 — Har power ka coefficient zero hona chahiye

KYA HAI. Ek polynomial (finite ya infinite) sabhi $x$ ke liye zero hota hai tabhi jab har coefficient zero ho. Toh har $k$ ke liye: $(k + 2) (k + 1) a_{k + 2} + a_{k} = 0.$

Naye coefficient ke liye solve karo: $a_{k + 2} = - \frac{a _{k}}{( k + 2 ) ( k + 1 )}$

Padho:

$a_{k + 2}$ — woh coefficient jo hum discover karne wale hain.
$a_{k}$ — ek coefficient do steps peeche, already known.
$- \frac{1}{( k + 2 ) ( k + 1 )}$ — shrink-and-flip factor: negative (kyunki $+ a_{k}$ hai), aur denominator badhta hai, toh terms jaldi chhote ho jaate hain.

Yeh ek recurrence relation hai — Recurrence relations dekho.

KyUN "DO STEPS PEECHE." Relation $k \to k + 2$ jump karta hai, toh evens sirf evens se baat karte hain aur odds sirf odds se. Do alag chains.

PICTURE. Ek ladder: har rung do rung neeche se banta hai, ek even ladder aur ek odd ladder mein split.

Step 6 — $a_{0}$ aur $a_{1}$ FREE hain — do loose ends

KYA HAI. Recurrence ko ek $a_{k}$ do peeche chahiye. Lekin $a_{0}$ ke do peeche kuch nahi hai, aur $a_{1}$ ke bhi nahi. Toh machine kabhi fix nahi karti $a_{0}$ ya $a_{1}$ ko — unhe hum choose karte hain.

EXACTLY DO KyUN. Ek second-order ODE (highest derivative $y^{''}$ ) ko hamesha do constants of integration chahiye. Yahan woh do ladder-bottoms ke roop mein appear hote hain: $a_{0}$ even ladder start karta hai, $a_{1}$ odd ladder start karta hai. Yeh Existence and uniqueness for linear ODEs se guaranteed hai.

PICTURE. Do ladders do free feet $a_{0}$ aur $a_{1}$ par khade hain.

Step 7 — Ladders crank karo aur answer pehchano

KYA HAI. Recurrence $a_{k + 2} = - \frac{a _{k}}{( k + 2 ) ( k + 1 )}$ feed karo.

Even chain ( $a_{0}$ se): $a_{2} = - \frac{a _{0}}{2 \cdot 1} = - \frac{a _{0}}{2 !}, a_{4} = - \frac{a _{2}}{4 \cdot 3} = + \frac{a _{0}}{4 !}, a_{6} = - \frac{a _{0}}{6 !}, \dots$

Odd chain ( $a_{1}$ se): $a_{3} = - \frac{a _{1}}{3 \cdot 2} = - \frac{a _{1}}{3 !}, a_{5} = - \frac{a _{3}}{5 \cdot 4} = + \frac{a _{1}}{5 !}, \dots$

Kis foot par khade hain uske hisaab se collect karo: $y = a_{0} = c o s x (1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots) + a_{1} = s i n x (x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots) .$

Alternating signs recurrence mein minus se aate hain, har step flip karte hain.
Factorials growing denominator $(k + 2) (k + 1)$ ke multiply hone se aate hain.

KyUN YEH KAMIYABI HAI. Humne kabhi $cos$ ya $sin$ assume nahi kiya tha. Equation ne unhe khud manufacture kiya, coefficient by coefficient. Yeh directly Taylor series and analyticity se connect hota hai — woh famous series exactly wahi hain jo ODE force karti hai. (Ek aise equation ke liye jiska series jaldi ruk jaata hai aur polynomial deta hai, Legendre's equation and Legendre polynomials dekho.)

PICTURE. Do collected columns, $cos x$ (blue) aur $sin x$ (pink) ke roop mein revealed.

Step 8 — Degenerate edge case: agar recurrence numerator zero ho jaaye toh?

KYA HAI. Hamare equation $y^{''} + y = 0$ mein factor hamesha $- \frac{1}{( k + 2 ) ( k + 1 )}$ hota hai, jo kabhi zero nahi hota — toh dono ladders forever chalte hain (do genuine infinite series). Lekin tumhe doosra case bhi pata hona chahiye. Ek Legendre-type equation mein recurrence aisa dikhta hai: $a_{k + 2} = \frac{( k + 3 ) ( k - 2 )}{( k + 2 ) ( k + 1 )} a_{k},$ aur $k = 2$ par numerator $(k - 2) = 0$ ho jaata hai, toh $a_{4} = 0$ aur poori even ladder uske aage collapse ho jaati hai. Woh branch ek finite polynomial ban jaata hai.

KyUN MATTER KARTA HAI. Jab bhi ek numerator factor kisi $k$ ke liye vanish ho sakta hai, woh ladder terminate ho jaati hai. Yahi mechanism Legendre aur Hermite polynomials ke peeche hai — series ko rokne ke liye engineer kiya jaata hai.

PICTURE. Do ladders side by side: haari (endless) vs. ek terminating wali jahan ek rung snap karta hai.

Ek-picture summary

Sab kuch — guess, differentiate, substitute, re-index, coefficients zero karo, do ladders crank karo, $cos$ aur $sin$ harvest karo — ek single flow mein compress kiya.

Recall Feynman retelling — isse bina notation ke ek dost ko bolo

Hum ek aisi curve $y$ chahte the jo "tumhari curviness tumhari height ki negative hai" — bas yahi kehti hai $y^{''} + y = 0$ . Hum koi normal function nahi jaante jo yeh kare, toh humne guess kiya ki curve ek giant polynomial hai unknown weights $a_{0}, a_{1}, a_{2}, \dots$ ke saath. Ek polynomial ko differentiate karna bas saare weights ko slide karta hai aur scale karta hai. Jab humne $y$ ko do baar slide kiya aur use khud se add kiya aur demand ki result har jagah flat-zero ho, toh humein ek rule mila: do aage wala har weight minus hai peeche wale weight ka, do badhte numbers se divide karke. Woh rule $a_{0}$ ya $a_{1}$ se peeche nahi ja sakta, toh woh do hum khud choose karte hain — ek second-order equation ke do dials. Crank ghoomane par, even dial $1, - \frac{1}{2}, + \frac{1}{24}, \dots$ print karta hai jo exactly $cos x$ hai, aur odd dial $x, - \frac{1}{6} x^{3}, \dots$ print karta hai jo $sin x$ hai. Equation ne hamare liye sine aur cosine build kar diye pure bookkeeping se. Aur agar chhoti factory rule mein kabhi "times zero" ka moment hota, toh ek ladder jaldi ruk jaati aur ek plain polynomial de deti — aise hi Legendre aur Hermite polynomials paida hote hain.