Separable ODEs — technique, implicit solutions

4.6.3 · Maths › Ordinary Differential Equations

Separable ODE KYA hoti hai?

Yeh kyun important hai: zyaadaatar ODEs closed form mein solve nahi hoti. Separable ODEs sabse friendly case hai — yahi pehla tool hai jo tum try karte ho. Agar yeh separate ho jaaye, toh basically sirf do integrals ke baad kaam khatam.

Technique KAISE kaam karti hai (derive karo, memorize mat karo)

Hum shuru karte hain $\frac{d y}{d x} = g (x) h (y) .$

Step 1 — $h (y)$ se divide karo (yeh maante hue ki $h (y) \neq = 0$ hai, yeh yaad rakhna!): $\frac{1}{h ( y )} \frac{d y}{d x} = g (x) .$

Step 2 — Dono sides ko $x$ ke respect mein integrate karo: $\int \frac{1}{h ( y )} \frac{d y}{d x} d x = \int g (x) d x .$

Step 3 — Left side pe substitution karo. Left wala integrand exactly chain-rule form mein hai: agar $H (y)$ , $1/ h (y)$ ka antiderivative hai, toh $\frac{d}{d x} H (y (x)) = \frac{1}{h ( y )} \frac{d y}{d x}$ . Toh left side ban jaata hai $\int \frac{d y}{h ( y )}$ : $\int \frac{d y}{h ( y )} = \int g (x) d x + C$

Figure — Separable ODEs — technique, implicit solutions

Implicit vs explicit solutions

Integrate karne ke baad tumhe $y$ aur $x$ ko relate karne wali ek equation milti hai, jaise $H (y) = G (x) + C$ . Kabhi kabhi tum $y$ ke liye solve kar sakte ho (explicit). Aksar cleanly nahi kar sakte — koi baat nahi, usse implicit solution ke roop mein chhod do.

Woh trap jo sabko lagta hai: lost solutions

Recall checkpoints

Recall Ek ODE separable kya banata hai, aur master formula kya hai?

RHS ko $g (x) h (y)$ ke roop mein factor hona chahiye. Phir $\int \frac{d y}{h ( y )} = \int g (x) d x + C$ .

Recall

$d y / d x$ ko fraction ki tarah treat karna yahan actually valid kyun hai? Yeh $\int \frac{1}{h ( y )} \frac{d y}{d x} d x = \int \frac{d y}{h ( y )}$ mein substitution rule (chain rule backwards) ka shorthand hai.

Recall

$h (y)$ se divide karke tumne possibly kya khoya, aur use recover kaise karte ho? Equilibrium solutions $y \equiv y_{0}$ jahan $h (y_{0}) = 0$ . Unhe divide karne se pehle constants ko alag se check karke recover karo.

Recall Ek 12-saal ke bachhe ko explain karo (Feynman)

Socho ek recipe hai jisme saare apple steps aur saare orange steps mix ho gaye. ODE ko separate karna unhe sort karne jaisa hai: saare apple ( $y$ ) instructions ek page pe daalo aur saare orange ( $x$ ) instructions doosre page pe. Phir tum har page ko integration se "undo" karte ho (slicing ka reverse), aur pages ko ek mystery number $C$ ke saath wapas tape karo jo tum ek starting fact se figure out karte ho jaise "shuru mein 3 apples the."

Flashcards

Ek first-order ODE separable kab hoti hai?

Jab

d y / d x

factor ho sake

g (x) h (y)

ke roop mein — pure-

x

times pure-

y

d y / d x = g (x) h (y)

ke liye master solution formula batao.

\int \frac{d y}{h ( y )} = \int g (x) d x + C

Separable ODE mein

d y / d x

ko fraction ki tarah "split" kyun kar sakte hain?

Yeh substitution/chain-rule step

\int \frac{1}{h ( y )} \frac{d y}{d x} d x = \int \frac{d y}{h ( y )}

ka shorthand hai.

Singular/equilibrium solution kya hai aur yeh kab aata hai?

Ek constant

y \equiv y_{0}

jahan

h (y_{0}) = 0

; jab

h (y)

se divide karte hain toh lost ho jaata hai.

d y / d x = x^{2} y

solve karo.

y = A e^{x^{3} /3}

d y / d x = \frac{2 x}{1 + 2 y}

solve karo (implicit).

y^{2} + y - x^{2} = C

Logistic

y^{'} = y (1 - y)

solve karo.

y = \frac{A e ^{x}}{1 + A e ^{x}}

plus equilibria

y = 0, y = 1

Ek first-order separable solution mein kitne arbitrary constants hote hain?

Exactly ek.

Kya ek implicit solution

F (x, y) = C

acceptable hai?

Haan — yeh ek valid solution hai chahe

y

isolate na ho sake.

SISC mnemonic kya hai?

Separate, Integrate both sides, Single constant, Check

h (y) = 0

for lost solutions.

Connections

First-Order ODEs — Overview
Integrating Factor & Linear ODEs (agla tool jab yeh separable nahi hoti)
Exact ODEs (implicit solutions yahan generalize hoti hain)
Substitution Method — Homogeneous ODEs (kuch non-separable ko separable mein convert karta hai)
Partial Fractions (Example 3 mein zaroori)
Initial Value Problems ( $C$ fix karna)
The Logistic Equation (Example 3 real world mein)