SymPy — symbolic algebra, calculus, ODE solving

5.4.20 · Coding › Scientific Computing (Python)

1. Symbolic computation kya hai?

x, y = symbols('x y')        # create two symbols
expr = x**2 + 2*x + 1        # an expression tree, NOT a number

TREE KYU? Kyunki x**2 + 2*x + 1 ki abhi koi value nahi hai — SymPy structure store karta hai taaki baad mein isse factor karke $(x + 1)^{2}$ bana sake.

2. Algebra: simplify, expand, factor, solve

Worked example Quadratic solve karo — derive karo, memorize mat karo

$x^{2} - 5 x + 6 = 0$ solve karo.

x = symbols('x')
solve(x**2 - 5*x + 6, x)     # -> [2, 3]

Yeh step kyu? SymPy internally $(x - 2) (x - 3)$ factor karta hai; ek product zero hota hai tab hi jab koi factor zero ho, isliye $x = 2, 3$ milta hai. Hum machine par trust karte hain lekin hum jaante hain kyun answer sahi hai.

Worked example Substitution + exact evaluation

e = (x + 1)**2
e.subs(x, sqrt(2))           # (sqrt(2) + 1)**2  -- still exact!
simplify(e.subs(x, sqrt(2))) # 2*sqrt(2) + 3

Yeh step kyu? .subs symbol ko swap karta hai; SymPy sqrt(2) ko symbolic rakhta hai. N(...) ya .evalf() sirf tab 5.828... dega jab hum decimal chahenge.

3. Calculus from first principles

Worked example Verify karo ki derivative IS the limit hai

x, h = symbols('x h')
f = x**2
limit((f.subs(x, x+h) - f)/h, h, 0)   # -> 2*x
diff(f, x)                            # -> 2*x   (same!)

Yeh step kyu? Yeh scratch-se-derivation hai: hum power rule assume nahi karte, hum isse limit definition se recover karte hain, phir confirm karte hain ki diff agree karta hai.

Formula Series (Taylor) — kyun exist karta hai

$f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$ series(f, x, 0, n) ise baar baar differentiate karke build karta hai. KYU: kisi point ke paas, ek smooth function polynomial jaisa dikhta hai — coefficients derivatives hote hain.

sin(x).series(x, 0, 6)   # x - x**3/6 + x**5/120 + O(x**6)

4. ODE solving — `dsolve`

Worked example Harmonic oscillator, derived

$y^{''} + y = 0$ solve karo.

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))

Yeh step kyu? $y = e^{r x}$ guess karo: tab $r^{2} e^{r x} + e^{r x} = 0 \Rightarrow r^{2} + 1 = 0 \Rightarrow r = \pm i$ . Complex roots $\pm i$ oscillations $cos x, sin x$ dete hain. dsolve exactly yahi characteristic-equation logic karta hai.

Worked example Initial conditions

dsolve(Eq(y(x).diff(x) - 2*y(x), 0), y(x),
       ics={y(0): 3})        # -> Eq(y(x), 3*exp(2*x))

Yeh step kyu? General solution $C_{1} e^{2 x}$ ; $y (0) = 3$ set karne se $C_{1} = 3$ pin hota hai. Forecast-then-verify: run karne se pehle $e^{2 x}$ growth predict karo — 2 seedha $y^{'} = 2 y$ se aaya.

Figure — SymPy — symbolic algebra, calculus, ODE solving

5. The 80/20 cheat-table

Goal	Call	Note
unknown banana	`x = symbols('x')`	assumptions ke liye `positive=True` etc. use karo
brackets kholna	`expand(e)`
ulta karna	`factor(e)`
saaf karna	`simplify(e)`	heuristic
plug in karna	`e.subs(x, v)`	exact
decimal lena	`e.evalf()` / `N(e)`
derivative	`diff(f, x)` / `diff(f,x,2)`	order extra arg se
integral	`integrate(f, x)` ya `(x,a,b)`
limit	`limit(f, x, 0)`
equation solve karna	`solve(Eq(l,r), x)`
ODE	`dsolve(Eq(...), y(x), ics=...)`	`Function` chahiye

Active Recall

Recall

symbols('x') kya deta hai aur sirf x = 5 kyun nahi? Ek unknown symbolic object jo algebra preserve karta hai. x=5 sab kuch numbers mein collapse kar deta hai, us structure ko destroy karke jo tum manipulate karna chahte the.

Recall

diff(x**2, x) scratch se sahi kaise confirm karoge? limit((( x+h)**2 - x**2)/h, h, 0) → 2*x, jo derivative ki limit definition se match karta hai.

Recall

(x+1)**2 == x**2+2*x+1 False kyun deta hai? == structural identity hai, math equality nahi. simplify(a-b)==0 use karo.

Recall

dsolve kya form return karta hai aur constants kaise fix karte hain? Ek general Eq(y(x), ...) C1, C2 ke saath; inhe pin karne ke liye ics={y(0): ...} supply karo.

Recall (Feynman, ek 12-saal ke bachche ko samjhao)

Normal calculators sirf numbers jaante hain — inhe √2 do aur yeh panic hokar 1.414... ban jaate hain. SymPy ek robot math teacher ki tarah hai jo "√2", "x", "π" symbols ko board par exactly likha rakhta hai aur wahi pencil rules follow karta hai jo tumne seekhe hain — brackets kholna, factor karna, differentiate karna — aur cheezein tabhi decimals mein badalta hai jab tum .evalf() se politely poochho. Toh yeh kabhi rounding se jhooth nahi bolta, aur tumhe ek formula deta hai jo tum padh sako.

Connections

NumPy — numerical arrays (numbers vs symbols: opposite philosophies)
Limits and the definition of the derivative
Taylor and Maclaurin Series
Ordinary Differential Equations — characteristic equation method
Lambdify — bridging SymPy to NumPy for plotting

SymPy kya manipulate karta hai jo NumPy nahi karta?

Exact symbolic objects (x, √2, π, 1/3) ko expression trees ki tarah, floats nahi.

Symbolic unknown x kaise banate hain?

x = symbols('x').

SymPy mein

(x + 1)^{2}

expand karo?

expand((x+1)**2) → x**2 + 2*x + 1.

x^{2} - 5 x + 6

factor karo?

factor(x**2-5*x+6) → (x-2)*(x-3).

x^{2} - 5 x + 6 = 0

solve karo?

solve(x**2-5*x+6, x) → [2, 3].

(x+1)² mein x=√2 substitute karo?

((x+1)**2).subs(x, sqrt(2)), exact rehta hai.

d/dx(x²) ki limit definition compute karo?

limit(((x+h)**2-x**2)/h, h, 0) → 2*x.

x² ka 0 se 1 tak definite integral?

integrate(x**2,(x,0,1)) → 1/3.

sin(x) ki order 6 tak Taylor series?

sin(x).series(x,0,6) → x - x**3/6 + x**5/120 + O(x**6).

y''+y=0 solve karo?

dsolve(Eq(y(x).diff(x,2)+y(x),0), y(x)) → C1*sin(x)+C2*cos(x).

Equality check ke liye == reliable kyun nahi hai?

Yeh structural identity test karta hai; simplify(a-b)==0 ya a.equals(b) use karo.

Symbolic result se decimal kaise lein?

.evalf() ya N(expr).

dsolve mein initial condition y(0)=3 kaise apply karein?

ics={y(0): 3} pass karo.