First-order linear ODEs — integrating factor method (derivation)

4.6.4 · Maths › Ordinary Differential Equations

FIRST-ORDER LINEAR ODE KYA HOTI HAI?

"Standard form" kyun important hai: neeche diye saare formulas yeh assume karte hain ki $\frac{d y}{d x}$ ka coefficient exactly $1$ hai. Agar tumhara $a (x) \frac{d y}{d x} + \dots$ hai, toh pehle puri equation ko $a (x)$ se divide karo.

INTEGRATING FACTOR DERIVE KAISE KAREIN (scratch se)

Hum poori equation ko kisi function $μ (x)$ se multiply karna chahte hain: $μ \frac{d y}{d x} + μ P y = μ Q .$

Apni left side $μ \frac{d y}{d x} + μ P y$ se term-by-term compare karo:

hamari left side	product rule
$μ \frac{d y}{d x}$	$μ \frac{d y}{d x}$ ✓ (pehle se match)
$μ P y$	$\frac{d μ}{d x} y$

Toh sirf yahi ek condition chahiye — $y$ ke coefficients match karein: $\frac{d μ}{d x} = μ P (x)$

Yeh khud $μ$ ke liye ek separable ODE hai! Ise solve karo: $\frac{d μ}{μ} = P d x \Rightarrow \int \frac{d μ}{μ} = \int P d x \Rightarrow ln ∣ μ ∣ = \int P d x .$

Exponentiate karo (constant drop karo — hume sirf ek kaam karne wala factor chahiye):

SOLUTION KAISE COMPLETE KAREIN

Multiply karne ke baad, equation ban jaati hai: $\frac{d}{d x} (μ y) = μ Q .$ Dono sides ko $x$ ke saath integrate karo: $μ y = \int μ Q d x + C .$

Figure — First-order linear ODEs — integrating factor method (derivation)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (answers cover karo)

Method apply karne se pehle $\frac{d y}{d x}$ ka coefficient KYA hona chahiye? → $1$
Left side kaunsa single derivative ban jaati hai? → $\frac{d}{d x} (μ y)$
Constant $C$ kahan jaata hai? → sirf final integration step par

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho do bachche hain, " $\frac{d y}{d x}$ -wala bachcha" aur " $P y$ -wala bachcha", jo haath nahi pakadna chahte isliye tum unhe saath nahi utha sakte. Integrating factor $μ$ ek special pair of mittens hai: jab sabne ek hi mittens pehne, woh jadu se ek bundle, $μ y$ , mein jud jaate hain, jise tum ek hi baar mein uthaa (integrate) sakte ho. Mittens aise chosen hain ki joining automatic ho — isliye $μ = e^{\int P}$ .

Flashcards

First-order linear ODE ki standard form?

\frac{d y}{d x} + P (x) y = Q (x)

Integrating factor ka formula?

μ (x) = e^{\int P (x) d x}

μ

par kaunsi condition left side ko single derivative banati hai?

\frac{d μ}{d x} = μ P (x)

μ

se multiply karne ke baad, LHS kaunse single derivative ke barabar hoti hai?

\frac{d}{d x} (μ y)

Linear ODE ka general solution?

y = \frac{1}{μ} [\int μ Q d x + C]

Agar

y^{'}

ka coefficient

1

nahi hai toh pehle kya karna chahiye?

Poori equation ko us coefficient se divide karo taaki standardize ho jaye

μ

compute karte waqt

+ C

kyun drop karte hain?

Yeh ek multiplicative

e^{C}

ban jaata hai jo cancel ho jaata hai; ek kaam karne wala factor kaafi hai

\frac{d y}{d x} + \frac{1}{x} y = \dots

ke liye

μ

μ = e^{\int \frac{1}{x} d x} = x

\frac{d y}{d x} + 2 y = e^{x}

ke liye integrating factor kya hai?

e^{2 x}

Integration constant kahan introduce hota hai?

Sirf final integration

μ y = \int μ Q d x + C

par

Connections

Separable ODEs — is derivation ke andar $μ$ solve karne ke liye use hota hai.
Product Rule — woh engine jo $μ \frac{d y}{d x} + μ P y = \frac{d}{d x} (μ y)$ possible banata hai.
Exact ODEs — integrating factors non-exact equations ko exact banane tak generalize hote hain.
Bernoulli Equations — substitution $v = y^{1 - n}$ se linear form mein reduce ho jaati hain.
Linear Constant-Coefficient ODEs — special case jahan $P, Q$ constant hain.