SymPy — symbolic algebra, calculus, ODE solving: Visual walkthrough

5.4.20 · D2 · Coding › Scientific Computing (Python) › SymPy — symbolic algebra, calculus, ODE solving

Prerequisite trails jahan tum wapas ja sakte ho: Limits and the definition of the derivative, Ordinary Differential Equations — characteristic equation method, Taylor and Maclaurin Series, aur home base the parent SymPy note.

Step 1 — Derivative hoti kya hai, drawn

KYA HAI. Kisi bhi equation se pehle, humein ek idea chahiye: derivative. Ek curve lo — ek height jo jaise tum daayein chalte ho, badlati jaati hai. Kisi point par derivative hai wahan curve ki steepness of the curve there — height kitni tezi se upar ya neeche ja rahi hai.

YEH TOOL KYU, DOOSRA KYU NAHI. Hum abhi $y^{'}$ aur $y^{''}$ symbol se milne waale hain. Agar hum yeh nail nahi karte ki geometrically unka matlab kya hai, toh yeh poora page sirf symbol-pushing ban jaayega. Isliye: pehle symbol earn karo. Hum slope-of-the-tangent-line picture use karte hain kyunki $y^{'}$ sirf wohi cheez measure karta hai, kuch aur nahi.

PICTURE. Figure dekho. White curve koi function $y (x)$ hai. Amber line ek hi point par curve ko just choomti hai — yeh tangent line hai. Uska tilt us point par $y^{'}$ hai.

Figure — SymPy — symbolic algebra, calculus, ODE solving

y^{''} = "rate of change of" \frac{d}{d x} the slope y^{'}

Ise zor se padho: " $y^{''}$ slope ke change ki rate hai." Isse zyada mysterious kuch nahi.

Step 2 — Equation $y^{''} + y = 0$ ko ek sentence ki tarah padhna

KYA HAI. Hum equation ko rewrite karte hain taaki iska matlab saaf sunai de:

y^{''} = - y

KYU. Symbols wali ek equation ek chupayi hui sentence hai. $y^{''}$ ko akela isolate karne ke liye rearrange karna ise ek physical rule mein badalta hai jise hum picture kar sakte hain. Hum $y$ ko daayein laate hain (dono sides se subtract karke) sirf isliye taaki "curving" baayein side pe akeli baithe.

PICTURE. Har symbol ek kaam kar raha hai:

how the curve bends y^{''} = opposite direction - current height y

Sentence: "tum jitne upar ho, exactly utna hi zero ki taraf wapas bend karo." Zyada upar → zyada neeche bend karo. Zero se neeche → upar bend karo. Zero par → bilkul bend mat karo. Figure dekho: amber arrows woh bending force $y^{''}$ hain, hamesha centre line ki taraf point karte hue, hamesha height ke barabar lambe.

Step 3 — Educated guess $y = e^{r x}$

KYA HAI. Hum ek candidate solution try karte hain:

y = e^{r x}

Yahan $e \approx 2.718$ ek fixed number hai, aur $r$ ek unknown number hai jise hum choose kar sakte hain.

YEH FUNCTION KYU, DOOSRA KYU NAHI. Humein ek aisi function chahiye jiska derivative khud jaisi lagti ho, kyunki equation $y$ aur $y^{''}$ compare karti hai — apples ko apples se milna chahiye. Exponential woh unique function hai jiske paas yeh magical property hai

\frac{d}{d x} e^{r x} = r e^{r x}

— differentiate karne par sirf $r$ se multiply ho jaata hai. Matlab $y$ , $y^{'}$ , $y^{''}$ sabhi $e^{r x}$ ki copies hain jo $r$ ke powers se scale hoti hain, toh equation $r$ mein simple algebra mein collapse ho jaayegi. Koi aur elementary function differentiation ke under apni shape nahi rakhti, isliye yeh the sahi tool hai.

PICTURE. Figure mein $e^{r x}$ aur uska derivative ek doosre ke upar dikh rahe hain, sirf ek vertical stretch factor $r$ se alag — same shape, rescaled.

take slope \frac{d}{d x} our guess e^{r x} = the stretch r same shape back e^{r x}

Step 4 — Plug in karo aur cancel karo: the characteristic equation

KYA HAI. Guess ko do baar differentiate karo aur $y^{''} + y = 0$ mein substitute karo.

y = e^{r x}, y^{'} = r e^{r x}, y^{''} = r^{2} e^{r x}

Substituting:

y^{''} r^{2} e^{r x} + y e^{r x} = 0

KYU. Yahi toh guess ka poora point tha: equation mein ab sirf $e^{r x}$ common factor ki tarah hai. Hum ise factor out karte hain:

e^{r x} (r^{2} + 1) = 0

Ab yahan key move hai. $e^{r x}$ kabhi zero nahi hota — ek exponential hamesha positive hoti hai, woh vanish nahi ho sakti. Toh product ke zero hone ka ek hi tarika hai — ki doosra factor zero ho:

r^{2} + 1 = 0

Yeh chhoti si equation characteristic equation kehlaayi jaati hai. Humne ek calculus problem ko algebra problem mein trade kar diya — yahi toh trick hai.

PICTURE. Figure mein exponential curve strictly zero se upar rehti hai har jagah (toh woh kabhi "zero" factor nahi ban sakti), jo $r^{2} + 1 = 0$ ko force karta hai.

never 0 e^{r x} \cdot must be 0 (r^{2} + 1) = 0

Step 5 — $r^{2} + 1 = 0$ solve karna: $i$ se milna

KYA HAI. $r$ ke liye solve karo:

r^{2} = - 1 ⟹ r = \pm i

$i$ symbol KYU. Humein ek aisa number chahiye jiska square $- 1$ ho. Koi ordinary number kaam nahi karta — kisi bhi real number ka square $\geq 0$ hota hai. Toh mathematicians ne ek naya number define kiya, jise $i$ kaha, ek simple rule se

i^{2} = - 1.

Bas itna hi $i$ ka matlab hai: "woh cheez jo square hokar $- 1$ deti hai." Hum ise sirf isliye introduce kar rahe hain kyunki equation ne maanga — algebra ek wall se takraayi, aur $i$ us wall ke paar ka darwaaza hai.

PICTURE. Figure mein number line ek plane mein bend ho gayi hai: real numbers left–right run karte hain, aur $i$ upar point karta hai, line se bahar. Do solutions $+ i$ aur $- i$ centre ke upar aur neeche baithe hain — mirror images.

Step 6 — $e^{i x}$ se real waves tak (Euler's bridge)

KYA HAI. Hamaare do solutions hain $y = e^{i x}$ aur $y = e^{- i x}$ . Lekin hum chahte hain ek real height — ek curve jo tum draw kar sako. Euler's identity imaginary exponential ko ordinary waves mein convert karti hai:

e^{i x} = cos x + i sin x

KYU. Hum $e^{i x}$ ke saath phas gaye hain, jo unplottable lagta hai. Euler's identity translator hai "imaginary ka exponential" se "sine aur cosine" tak. Yeh koi hawa se kheenchi gayi trick nahi hai — yeh seedha $e$ , $cos$ aur $sin$ ke Taylor and Maclaurin Series se nikalta hai (parent note ka series tool): teeno power series line up karo aur pattern exactly match karta hai.

PICTURE. Figure mein point $e^{i x}$ ek unit circle ke around travel karta dikhta hai jaise $x$ badhta jaata hai. Uski horizontal shadow $cos x$ trace karti hai; uski vertical shadow $sin x$ trace karti hai. Circular motion hai hi do oscillations right angles par — exactly wahi wobble jo humne Step 2 mein forecast ki thi.

e^{i x} = horizontal shadow cos x + i vertical shadow sin x

Step 7 — General solution assemble karna

KYA HAI. Do imaginary solutions milkar do real building blocks bante hain, $cos x$ aur $sin x$ . Kyunki equation linear hai (koi $y^{2}$ nahi, koi $y \cdot y^{'}$ nahi), solutions ka koi bhi weighted mix phir se ek solution hoga. Toh:

y (x) = kitna sine C_{1} sin x + kitna cosine C_{2} cos x

DO CONSTANTS KYU. Ek second-derivative equation ( $y^{''}$ ) ko ek unique curve pin down karne ke liye do facts chahiye — tum kahan se shuru karte ho ( $y (0)$ ) aur kitni tezi se shuru karte ho ( $y^{'} (0)$ ). Har fact ek constant "khaata" hai. Exactly isliye dsolve $C_{1}$ aur $C_{2}$ return karta hai: use abhi tumhari starting conditions pata nahi hain.

PICTURE. Figure mein family ke kai members overlay kiye hain — alag $C_{1}, C_{2}$ alag height aur phase ki waves dete hain, lekin har ek dolta hai, jo Step 2 ke spring forecast ko confirm karta hai.

x = symbols('x')
y = Function('y')
dsolve(Eq(y(x).diff(x, 2) + y(x), 0), y(x))
# -> Eq(y(x), C1*sin(x) + C2*cos(x))

Step 8 — Degenerate aur edge cases (koi gap mat chhodo)

KYA HAI. Agar equation thodi alag hoti toh? Hum neighbours check karte hain taaki koi scenario tumhe surprise na kare.

KYU. Samajh ka judge coverage hai: tumhe pata hona chahiye ki yeh equation waves kyun deti hai aur ek cousin kuch aur kyun deta hai.

Equation	Characteristic eqn	Roots $r$	Solution shape	Picture
$y^{''} + y = 0$	$r^{2} + 1 = 0$	$\pm i$ (imaginary)	$cos x, sin x$ — pure waves	Step 7
$y^{''} - y = 0$	$r^{2} - 1 = 0$	$\pm 1$ (real)	$e^{x}, e^{- x}$ — bhaag jaana / decay	neeche
$y^{''} = 0$	$r^{2} = 0$	$0, 0$ (repeated)	$1, x$ — seedhi line	neeche
$y^{'} - 2 y = 0$	$r - 2 = 0$	$2$ (single)	$e^{2 x}$ — growth, ek constant	parent §4

Characteristic equation ke andar ka sign sab decide karta hai: $+$ imaginary roots aur oscillation deta hai; $-$ real roots aur exponential growth/decay deta hai; zero par repeated root seedhi line deta hai. Figure mein teeno side by side draw hain.

Ek-picture summary

Yeh single figure saare aath steps compress karti hai: equation ka matlab (spring), exponential guess, algebra collapse, imaginary roots vertical axis par, Euler's circle, aur do real waves jo bahar aati hain.

Recall Feynman retelling — poora walkthrough simple shabdon mein

Humne ek curve se shuru kiya aur seekha ki $y^{'}$ uski steepness hai aur $y^{''}$ steepness kaise bend hoti hai. Equation $y^{''} + y = 0$ literally kehti hai "beech ki taraf utna hi wapas bend karo jitna tum door ho" — yeh ek spring hai, toh humne guess kiya ki answer dolta hai. Use find karne ke liye humne $y = e^{r x}$ try kiya, woh ek function jo slope lete waqt khud jaisi rehti hai, toh calculus simple algebra mein badal gayi: $r^{2} + 1 = 0$ . Iske liye ek aisa number chahiye jo square hokar $- 1$ de, toh $i$ aa gaya, diya $r = \pm i$ . Euler's identity ne phir kaha ki $e^{i x}$ sirf ek point hai jo ek circle ke around ghoom raha hai, jiske do shadows $cos x$ aur $sin x$ hain. Un do waves ko kisi bhi amount $C_{1}, C_{2}$ mein mix karo aur tumhare paas har possible solution hai — exactly wohi jo dsolve print karta hai. Ek sign change karo aur roots real ho jaate hain aur wobble runaway growth ban jaata hai. Yahi poori kahani hai: ek spring, ek clever guess, ek darwaaza jiska naam $i$ hai, aur do waves.

Recall

dsolve $y^{''} + y = 0$ ke liye do constants kyun return karta hai lekin $y^{'} - 2 y = 0$ ke liye sirf ek kyun? Highest derivative ki order equals the number of independent starting facts jo tumhe supply karni padti hain, isliye arbitrary constants ki bhi utni hi sankhya: second order → $C_{1}, C_{2}$ ; first order → sirf $C_{1}$ .

Recall

$e^{r x}$ ko factor out karna legal aur useful kyun hai? $e^{r x}$ kabhi zero nahi hota, toh isse divide karne (ya factor karne) par koi solutions na loss hote hain na invent — aur yeh calculus ko strip kar deta hai, purely algebra $r^{2} + 1 = 0$ bachta hai.

Connections

Limits and the definition of the derivative — Step 1 ka slope idea, rigorous banaya hua.
Ordinary Differential Equations — characteristic equation method — woh hand method jo dsolve automate karta hai.
Taylor and Maclaurin Series — jahan se Euler's $e^{i x} = cos x + i sin x$ (Step 6) aata hai.
Lambdify — bridging SymPy to NumPy for plotting — $C_{1} sin x + C_{2} cos x$ ko numbers mein badlo plot karne ke liye.
NumPy — numerical arrays — woh numeric counterpart jo sirf decimals dekhta.