SymPy — symbolic algebra, calculus, ODE solving

1. What is symbolic computation?

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

WHY a tree? Because x**2 + 2*x + 1 has no value yet — SymPy stores structure so it can later factor it into $(x+1)^2$ .

2. Algebra: simplify, expand, factor, solve

Worked example Solve a quadratic — derive, don't memorize

Solve $x^2 - 5x + 6 = 0$ .

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

Why this step? SymPy factors $(x-2)(x-3)$ internally; a product is zero iff a factor is zero, giving $x=2,3$ . We trust the machine but we know why the answer is right.

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

Why this step? .subs swaps the symbol; SymPy keeps sqrt(2) symbolic. N(...) or .evalf() would give 5.828... only when we want a decimal.

3. Calculus from first principles

Worked example Verify the derivative IS the limit

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!)

Why this step? This is derivation-from-scratch: we don't assume the power rule, we recover it from the limit definition, then confirm diff agrees.

Worked example Definite integral

integrate(x**2, (x, 0, 1))    # -> 1/3   (exact rational!)

Why this step? $\int_0^1 x^2 dx = [x^3/3]_0^1 = 1/3$ . Numpy would give 0.3333...; SymPy gives the exact 1/3.

Formula Series (Taylor) — why it exists

$f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$ series(f, x, 0, n) builds this by repeatedly differentiating. WHY: near a point, a smooth function looks like a polynomial — the coefficients are derivatives.

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

Solve $y'' + y = 0$ .

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))

Why this step? Guess $y=e^{rx}$ : then $r^2 e^{rx} + e^{rx}=0 \Rightarrow r^2+1=0 \Rightarrow r=\pm i$ . Complex roots $\pm i$ give oscillations $\cos x,\sin x$ . dsolve does exactly this characteristic-equation logic.

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))

Why this step? General solution $C_1 e^{2x}$ ; setting $y(0)=3$ pins $C_1=3$ . Forecast-then-verify: predict growth $e^{2x}$ before running — the 2 came straight from $y'=2y$ .

Figure — SymPy — symbolic algebra, calculus, ODE solving

5. The 80/20 cheat-table

Goal	Call	Note
make unknown	`x = symbols('x')`	use `positive=True` etc. for assumptions
open brackets	`expand(e)`
reverse	`factor(e)`
tidy	`simplify(e)`	heuristic
plug in	`e.subs(x, v)`	exact
decimal	`e.evalf()` / `N(e)`
derivative	`diff(f, x)` / `diff(f,x,2)`	order via extra arg
integral	`integrate(f, x)` or `(x,a,b)`
limit	`limit(f, x, 0)`
solve eqn	`solve(Eq(l,r), x)`
ODE	`dsolve(Eq(...), y(x), ics=...)`	needs `Function`

Active Recall

Recall What does

symbols('x') give you and why not just x = 5? An unknown symbolic object that preserves algebra. x=5 collapses everything to numbers, destroying the structure you wanted to manipulate.

Recall How would you confirm

diff(x**2, x) is correct from scratch? limit((( x+h)**2 - x**2)/h, h, 0) → 2*x, matching the limit definition of the derivative.

Recall Why does

(x+1)**2 == x**2+2*x+1 give False? == is structural identity, not math equality. Use simplify(a-b)==0.

Recall What form does

dsolve return and how do you fix constants? A general Eq(y(x), ...) with C1, C2; supply ics={y(0): ...} to pin them.

Recall (Feynman, explain to a 12-year-old)

Normal calculators only know numbers — give them √2 and they panic into 1.414.... SymPy is like a robot math teacher who keeps the symbols "√2", "x", "π" written exactly on the board and follows the same pencil rules you learned — open brackets, factor, differentiate — and only turns things into decimals when you ask politely with .evalf(). So it never lies by rounding, and it hands you a formula you can read.

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

What does SymPy manipulate that NumPy does not?

Exact symbolic objects (x, √2, π, 1/3) as expression trees, not floats.

How do you create a symbolic unknown x?

x = symbols('x').

Expand

(x+1)^2

in SymPy?

expand((x+1)**2) → x**2 + 2*x + 1.

Factor

x^2-5x+6

factor(x**2-5*x+6) → (x-2)*(x-3).

Solve

x^2-5x+6=0

solve(x**2-5*x+6, x) → [2, 3].

Substitute x=√2 into (x+1)²?

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

Compute the limit definition of d/dx(x²)?

limit(((x+h)**2-x**2)/h, h, 0) → 2*x.

Definite integral of x² from 0 to 1?

integrate(x**2,(x,0,1)) → 1/3.

Taylor series of sin(x) to order 6?

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

Solve y''+y=0?

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

Why is == unreliable for checking equality?

It tests structural identity; use simplify(a-b)==0 or a.equals(b).

How to get a decimal from a symbolic result?

.evalf() or N(expr).

How to apply initial condition y(0)=3 in dsolve?

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, SymPy ka core idea simple hai: normal calculator aur NumPy sirf numbers ke saath khelte hain — tum √2 doge to woh turant 1.414... bana deta hai aur asli cheez (exactness) kho jaati hai. SymPy ek robot maths teacher ki tarah hai jo x, sqrt(2), pi, 1/3 ko exactly waise hi rakhta hai jaise tum copy par likhte ho. Isliye pehla rule: variable ko x = symbols('x') se banao, kabhi x = 5 mat karo — warna algebra khatam.

Algebra ke 4 main moves hain: expand (brackets kholo), factor (ulta), simplify (saaf karo), aur solve (equation ka root nikaalo). Yaad rakho solve(expr, x) ka matlab expr = 0 hota hai, kyunki har equation $a=b$ ko $a-b=0$ likha ja sakta hai. Aur ek bada trap: == se equality mat check karna — woh structure compare karta hai, isliye simplify(a-b)==0 use karo.

Calculus mein diff derivative deta hai, integrate integral, limit limit, aur series Taylor expansion. Best part — tum khud derivation-from-scratch kar sakte ho: limit(((x+h)**2 - x**2)/h, h, 0) chalao aur dekho 2*x aata hai, bilkul power rule jaisa. Yeh confidence deta hai ki machine jhooth nahi bol rahi.

ODE ke liye dsolve hai. Pehle y = Function('y') banao, phir y(x).diff(x,2) likho. $y''+y=0$ ka answer C1*sin(x)+C2*cos(x) aata hai — yeh characteristic equation $r^2+1=0$ se nikalta hai. Initial condition dene ke liye ics={y(0):3} pass karo, taaki constants fix ho jaayein. Exam aur research dono mein yeh tumhe formula wapas deta hai, sirf number nahi — isi liye SymPy itna powerful hai.

SymPy — symbolic algebra, calculus, ODE solving

1. What is symbolic computation?

2. Algebra: simplify, expand, factor, solve

3. Calculus from first principles

4. ODE solving — `dsolve`

5. The 80/20 cheat-table

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Scientific Computing (Python)

Connections