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

4.6.4 · D1 · Maths › Ordinary Differential Equations › First-order linear ODEs — integrating factor method (derivat

Isse pehle ki tum parent derivation follow kar sako, us page ka har symbol tumhare liye kuch concrete maana rakhna chahiye. Neeche hum har ek ko bilkul zero se build karte hain, us order mein jisme wo ek doosre par depend karte hain — koi bhi symbol apne section se pehle use nahi kiya jaata.

1. Symbols $x$ aur $y$, aur "$x$ ka function" kya hota hai

Ek curve ki page par khinchi hui tasveer socho. Horizontal axis $x$ hai, vertical axis $y$ hai. Jaise tum $x$ slide karte ho, tumhari ungli curve par upar-neeche trace karti hai — wahi trace function hai.

Topic ko iske kyon zaroorat hai: ek ODE ek aisi equation hai jiska unknown poori curve $y(x)$ hoti hai, na ki koi single number. Ise solve karna matlab hai ki kaunsi curve fit hoti hai woh dhundhna.

2. Derivative $\frac{dy}{dx}$ — curve ki steepness

Yeh genuinely nayi notation ka pehla piece hai. $\frac{dy}{dx}$ ko "dee-y dee-x" padhkar suno, aur ise ek single object ki tarah samjho jiska matlab hai "$x$ ke move karne par $y$ kitni tezi se badhti hai."

"Derivative" kyun kehte hain, sirf "slope of a line" kyun nahi? Ek curve ki steepness point-to-point badlti hai. Derivative woh tool hai jo tumhe har $x$ par alag-alag steepness deta hai — yeh khud $x$ ka ek function hai. Isliye ek ODE curve ke har point par hold kar sakta hai.

3. Coefficients $P(x)$ aur $Q(x)$

Ek machine par do extra dials socho, har ek par ek number likha ho jo $x$ slide karte waqt badlta hai. $x=2$ par, shayad $P=3$ aur $Q=e^2$ ho; $x=5$ par, alag numbers. Ye tumhe diye jaate hain — tumhara kaam $y$ dhundhna hai.

Ab parent equation poori tarah yun likhti hai, $\frac{dy}{dx}+P(x)y=Q(x),$ aur isme har symbol — $\frac{dy}{dx}$ (section 2), $y$ (section 1), $P,Q$ (yahan) — ab build ho chuka hai.

Topic ko "$x$ only" restriction ki kyun zaroorat hai: agar $P$ mein $y$ hota (maan lo $P=y^2$), toh equation nonlinear ho jaati aur product-rule trick kaam nahi karti. Poora method is baat par depend karta hai ki $P$ aur $Q$, $y$ se blind hoon.

4. "Linear" ka matlab kya hai — exact power rule

Figure ka left side: allowed shapes — $y$, $\frac{dy}{dx}$, $x$ ke function se multiply, add kiye gaye. Right side: forbidden shapes — $y^2$, $y\frac{dy}{dx}$, $\sin y$.

Topic ko iske kyon zaroorat hai: "linear" precisely woh class hai jiske liye poora left side baad mein ek single derivative mein absorb ho sakta hai. Yeh entry ticket hai.

5. Integral sign $\int$ — derivative ko undo karna

Figure round trip dikhati hai: ek curve $\to$ (derivative) $\to$ uski slope $\to$ (integral) $\to$ curve par wapas, plus ek possible vertical shift (accent-red arrow) — kyunki ek curve ko upar ya neeche slide karne se uski slope nahi badlti.

Topic ko iske kyon zaroorat hai: poore method ka final move "dono sides integrate karo" hai. Woh $+C$ precisely woh freedom hai jo ek ODE ko curves ki poori family describe karne deti hai, jo sirf ek initial condition se pin down hoti hai. Separable ODEs dekho jahan yeh backwards-machine uske baad aane wali derivation ke andar use hoti hai.

6. Exponential $e^{(\cdots )}$ aur uska inverse $\ln$

Figure: $e^x$ aur $\ln x$ line $y=x$ (accent red) ke paas mirror images hain. Ek ko doosre mein feed karne par tum wahan wapas pahunchte ho jahan se shuru kiya tha — wahi mirror "inverse" ka matlab hai.

Topic ko iske kyon zaroorat hai: woh special function jo hum build karne wale hain woh exponential nikalta hai, aur ise simplify karne ka matlab hamesha ek $e$ ko ek $\ln$ se cancel karna hota hai. Iske bina tum $e^{\ln |x|}$ par atke rehte ho aur finish nahi kar sakte.

7. Product rule — poori trick ka engine

Yeh sabse important prerequisite hai. Product Rule ko tab tak study karo jab tak yeh automatic na ho jaaye.

8. Special multiplier $\mu (x)$ build karna

Upar diya gaya sab kuch ab hume us ek nayi object define karne deta hai jise parent note integrating factor kehta hai.

Standard equation $\frac{dy}{dx}+Py=Q$ ko ek as-yet-unknown $\mu$ se multiply karo: $\mu \frac{dy}{dx}+\mu Py=\mu Q.$ Ab section 7 ke product rule mein $u=\mu$ aur $v=y$ set karo: $\frac{d}{dx}(\mu y)=\mu \frac{dy}{dx}+\frac{d\mu }{dx}y.$

$\frac{d\mu }{dx}=\mu P$ ko separation of variables se solve karna

Yeh chhoti si equation khud solve ho sakti hai — yeh $\mu$ ke liye ek separable ODE hai. Yahan "variables separate karna" ka matlab hai: $\mu$ se related sab kuch ek side par aur $x$ se related sab kuch doosri side par laao.

Step 1 — $\mu$ se divide karo (KYA & KYUN): right se $\mu$ hatane ke liye, dono sides ko $\mu$ se divide karo, saara $\mu$-stuff left mein move karo: $\frac{1}{\mu }\frac{d\mu }{dx}=P(x).$

Step 2 — $dx$ se multiply karo (KYA & KYUN): har side ko apne integral ke neeche line up karne ke liye, $\mu$-differential left par aur $x$-differential right par gather karo: $\frac{d\mu }{\mu }=P(x)dx.$

Step 3 — dono sides integrate karo (KYA & KYUN): ab har side ek pure "reverse-slope" problem hai (section 5). Left woh classic integral hai jiska answer logarithm hai, $\int \frac{d\mu }{\mu }=\ln |\mu |$; right $\int Pdx$ hai: $\ln |\mu |=\int P(x)dx+C_0.$ Isliye $P$ ko integrate karne se exponent nikalta hai — $\int Pdx$ literally wahi hai jo right-hand side se nikalta hai, aur woh ek logarithm ke andar land karta hai.

Step 4 — logarithm ko undo karo (KYA & KYUN): $\mu$ ko $\ln$ ke andar se azaad karne ke liye, dono sides par uska inverse $e^{(\cdots )}$ (section 6) apply karo: $|\mu |=e^{\int Pdx+C_0}=e^{C_0}{e}^{\int Pdx}.$

Topic ko iske kyon zaroorat hai: product rule ko reverse mein chalta pehchaane bina, collapse $\mu \frac{dy}{dx}+\mu Py=\frac{d}{dx}(\mu y)$ magic jaisi lagti hai. Iske saath, yeh inevitable hai.

Foundations topic ko kaise feed karte hain

Ise top-down padho: variable aur derivative ke notions dono "linear" ke matlab aur product rule ko feed karte hain; product rule plus exponential plus integration, integrating factor $\mu$ build karte hain; aur $\mu$ integration ke saath milkar method deliver karta hai.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi answer fuzzy lage, parent note kholne se pehle uska section upar se revisit karo.

Connections

• Product Rule — woh engine jo $\mu \frac{dy}{dx}+\mu Py$ ko $\frac{d}{dx}(\mu y)$ mein badalta hai.
• Separable ODEs — woh method jo derivation ke andar $\frac{d\mu }{dx}=\mu P$ solve karne ke liye use hota hai.
• Exact ODEs — integrating factors wahan bhi equations ko exact banane ke liye appear karte hain.
• Bernoulli Equations — ek substitution ke baad linear (yeh topic) ban jaati hain.
• Linear Constant-Coefficient ODEs — woh special case jahan $P,Q$ constants hain.