Foundations — Integrating factors for non-exact equations

4.6.7 · D1 · Maths › Ordinary Differential Equations › Non-exact equations ke liye Integrating factors

Parent note use karne se pehle, tumhare paas uski language ka har ek piece hona chahiye. Hum har symbol ko kuch nahin se build karte hain, use ek picture se anchor karte hain, aur kehte hain kyun is topic ko yeh chahiye. Upar se neeche padho — har item upar wale par lean karta hai.

1. Do variables ke functions: $M(x,y)$, $N(x,y)$, $F(x,y)$

Ek flat map ki picture karo ($x$–$y$ plane zameen par rakhhi hui). Har point par jo tum khade ho, function tumhe ek number deta hai — sochो sea level ke upar ki height. Toh $F$ ek landscape hai: plane ke upar ek hill.

• $x$ = kitna east tum khade ho (horizontal axis).
• $y$ = kitna north tum khade ho (doosra horizontal axis).
• $F(x,y)$ = hill ki height seedha tumhare upar.

Kyun topic ko yeh chahiye. Puri method ek hidden height-function $F$ dhundhti hai jiske contour lines ODE ke solutions hain. Aur $M$, $N$ khud do-variable functions hain — woh plane ke har point par ek value lete hain.

2. Contour lines aur "$F=C$"

Ek real hiking map par, contour lines equal height ki jagahon ko join karti hain. Ek contour ke saath chalo toh tum kabhi upar ya neeche nahin jaate.

Figure mein green loops curves $F=1$, $F=2$, $F=3$ hain. ODE solve karne ka poora point yahi loops naam karna hai — answer "$F(x,y)=C$" matlab hai "solution woh contour hai jis par tum shuru huey the."

Kyun topic ko yeh chahiye. Parent mein boxed answers (jaise $x^{3}y+\frac{1}{2}{x}^2{y}^2=C$) exactly "$F$ ka yeh contour" hain. Contours nahin, solution nahin.

3. Differential pieces $dx$, $dy$ aur form $Mdx+Ndy$

Ek point par yellow tiny arrow $(dx,dy)$ dekho. Rule use score karta hai: $M$ eastward part ko weight karta hai, $N$ northward part ko weight karta hai. Ek aisi direction chunna jo $0$ score kare tumhe level curve par rakhti hai.

Kyun topic ko yeh chahiye. Parent har first-order ODE ko is differential form $Mdx+Ndy=0$ mein likhta hai. Woh single line woh object hai jise hum test karte hain, multiply karte hain, aur solve karte hain.

4. Partial derivatives $\frac{\partial F}{\partial x}$, $F_x$, $M_y$, $N_x$

Word partial matlab "ek direction ek baar." $y$ ko constant ki tarah freeze karo, phir $x$ ke baaki reh gaye one-variable function ko normal tarah differentiate karo.

• Blue slice: hill ko east ki taraf wall se kaato; us slice ki tilt $F_x$ hai.
• Red slice: north ki taraf wall se kaato; uski tilt $F_y$ hai.

Kyun topic ko yeh chahiye. Exactness test $M_y=N_x$, integrating-factor PDE, aur $F$ ki recovery sab partial derivatives se bane hain. Yeh page ka sabse zyada use hone wala single tool hai.

5. Total differential $dF=F_{x}dx+F_{y}dy$

Yeh bas "rise = slope × run" hai, dono directions mein kiya gaya aur add kiya gaya. Yeh hill picture aur walking-rule picture ke beech ka bridge hai.

Kyun topic ko yeh chahiye. Yahi exact ki definition hai: ek equation exact tab hai jab uska left side kisi ka total differential ho.

6. Kyun $M_y=N_x$ exactness test karta hai (Clairaut)

Agar $M=F_x$ aur $N=F_y$, tab $M_y=F_{xy}$ aur $N_x=F_{yx}$, aur Clairaut in dono ko equal force karta hai. Toh:

Kyun topic ko yeh chahiye. Yeh cheap check hai jo humein bataata hai ki koi $F$ exist bhi karta hai ya nahin, isse pehle ki hum use dhundhein — aur woh equation jise integrating factor true force karne ke liye design kiya gaya hai. (Vector-field readers ke liye: yeh curl-free / conservative condition hai.)

7. Exponential aur integral: $\exp$, $\int$, aur $\frac{{\mu }^{\prime }}{\mu }$

Kyun topic ko yeh chahiye. Do boxed formulas $\mu (x)=\exp \int \frac{M_y-N_x}{N}dx$ aur $\mu (y)=\exp \int \frac{N_x-M_y}{M}dy$ yahi idea hain. Yahi $e^{\int Pdx}$ Linear first-order ODEs mein bhi aata hai — yeh wahi trick hai.

8. Partial integration aur $\varphi (y)$ se $F$ ki recovery

Phir tum apne candidate $F$ ko $y$ mein differentiate karte ho, use $N$ se match karte ho, aur yahi ${\varphi }^{\prime }(y)$ aur isliye $\varphi (y)$ ko pin down karta hai.

Kyun topic ko yeh chahiye. Yeh parent ki procedure ka step 5 hai — tum actually potential $F$ kaise produce karte ho jab equation exact ho jaaye. $\varphi (y)$ bhoolna listed classic mistakes mein se ek hai.

Yeh sab topic ko kaise feed karta hai

Equipment checklist

Self-test: reveal karne se pehle kya tum har ek ka jawab de sakte ho?

Connections

• Integrating factors for non-exact equations — woh parent jiske liye yeh page tumhe tayaar karta hai
• Exact differential equations
• Total differentials and potential functions
• Mixed partial derivatives (Clairaut's theorem)
• Linear first-order ODEs
• Separable equations
• Conservative vector fields