Non-homogeneous — variation of parameters: Foundations

4.6.15 · D1 · Maths › Ordinary Differential Equations › Non-homogeneous — variation of parameters

Is page pe assume kiya gaya hai ki aapne kuch nahi dekha. Hum page pe har mark ka naam lete hain, uska picture dikhate hain, aur batate hain ki topic uske bina kyun exist nahi kar sakta. Upar se neeche padho; har item usse upar wale pe lean karta hai.

1. Function aur uska naam $y (x)$

Picture. Figure s01 dekho. Horizontal axis ( $x$ ) pe ek pointer slide karo. Us jagah ke upar curve ki height value $y$ hai. Poori curve hi function hai.

Topic ko yeh kyun chahiye. Ek "ordinary differential equation" (ODE) ek aisi puzzle hai jiska answer ek poori function $y (x)$ hoti hai, koi single number nahi. Toh pehle hume comfortable hona hoga ki unknown ek curve hai.

2. Prime — ek derivative $y^{'}$

Picture. Figure s02 dekho. Kisi bhi point pe chhoti tangent line curve ko touch karti hai; uska tilt $y^{'}$ hai. Ab us tilt ko dekho jab aap right slide karte ho: ek curve jo upar ki taraf bend karti hai (valley shape) mein tangent pehle down tilts karta hai, flat hota hai, phir up tilts karta hai — tangent line counter-clockwise rotate karti hai. Us rotation ki speed $y^{''}$ hai. Bada $y^{''}$ matlab tangent zyada tez swing karta hai, yaani curve zyada sharply bend karti hai.

Topic ko yeh kyun chahiye. Poora equation $y^{''} + p y^{'} + q y = g$ sirf $y, y^{'}, y^{''}$ se bana hai. Prime ke bina hum problem likh bhi nahi sakte.

3. Equation $y^{''} + p y^{'} + q y = g$

"Second-order" isliye kyunki sabse bada prime $y^{''}$ hai (do primes).
"Linear" isliye kyunki $y, y^{'}, y^{''}$ sirf first power mein aate hain — koi $y^{2}$ nahi, koi $sin y$ nahi.

Topic ko yeh kyun chahiye. Yahi woh equation hai jo variation of parameters solve karta hai. Left side machine hai; right side woh hai jo hum print karwana chahte hain.

4. Homogeneous vs non-homogeneous, aur $g (x)$

Picture. Figure s03 dekho. Left panel: ek pendulum freely swing karta hua — homogeneous. Right panel: ek haath use periodically nudge kar raha hai — woh nudge $g (x)$ hai, non-homogeneous.

Related, faster-but-narrower tool: Method of Undetermined Coefficients sirf special pushes handle karta hai.

5. Do building-block solutions $y_{1}, y_{2}$

"Do" kyun? Second-order $\Rightarrow$ do independent choices (jaise starting position aur starting speed choose karna). Yeh nikalna Second-order linear homogeneous ODE ka kaam hai, aksar Reduction of Order ke zariye jab aap ek jaante ho.

Topic ko yeh kyun chahiye. Poora answer $y_{1}, y_{2}$ se bana hai. Yeh raw lumber hain.

6. Constants vs functions: $c_{1}, c_{2} \to u_{1} (x), u_{2} (x)$

Picture. Figure s04. Upar: do knobs $c_{1}, c_{2}$ frozen. Neeche: wohi knobs, lekin ab haath unhe $x$ badhne ke sath ghuma rahe hain — "parameters ki variation."

7. Determinant bars aur Wronskian $W$

Plain meaning. $W$ ek "fairness scale" hai. Agar $W \neq = 0$ , $y_{1}$ aur $y_{2}$ sach mein alag building blocks hain; agar $W = 0$ , ek secretly doosre ki copy hai aur method jam ho jaata hai (zero se divide ho jaata). Poori baat: Wronskian.

Topic ko yeh kyun chahiye. $W$ final formula ke denominator mein hai — jab hum strengths solve karte hain toh yeh literally divisor hai.

8. Do equations do unknowns solve karna: Cramer's Rule

Setup se ek chhota system milta hai: ${u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0 u_{1}^{'} y_{1}^{'} + u_{2}^{'} y_{2}^{'} = g$

Yahan do unknowns $u_{1}^{'}$ aur $u_{2}^{'}$ hain (strengths kitni tez change ho rahi hain).

$u_{1}^{'}$ pe minus koi mystery nahi hai — yeh determinant ke $- b c$ se aata hai jab right side $(0, g)$ pehle column mein aata hai.

9. Integral sign $\int$

Topic ko yeh kyun chahiye. Cramer's rule se rates $u_{1}^{'}, u_{2}^{'}$ milte hain. Actual strengths $u_{1}, u_{2}$ nikalne ke liye hume prime undo karna hoga — yahi exactly integration hai: $u_{1} = \int u_{1}^{'} d x, u_{2} = \int u_{2}^{'} d x .$

10. Standard form (leading-coefficient trap)

Sab kuch topic ko kaise feed karta hai

Ise ek build order ki tarah padho — neeche har stage tabhi sense banata hai jab usse upar wale stages haath mein hoon:

Function $y (x)$ (§1) hume kuch differentiate karne layak deta hai.
Primes $y^{'}, y^{''}$ (§2) hume rates aur curvature likhne dete hain.
Saath mein woh second-order linear ODE (§3) banate hain.
Push alag karna homogeneous vs $g$ (§4) distinguish karta hai.
Homogeneous side do building blocks $y_{1}, y_{2}$ (§5) deta hai.
Hum constants ko functions $u_{1}, u_{2}$ se swap karte hain (§6) push tak pahunchne ke liye.
Blocks ki independence Wronskian $W$ (§7) se measure hoti hai.
Steps 6–7 ek 2×2 system feed karte hain jo Cramer's Rule (§8) se solve hota hai, $u_{1}^{'}, u_{2}^{'}$ milte hain.
Integrate karna (§9) un rates ko $u_{1}, u_{2}$ mein turn karta hai — particular solution.
Standard form (§10) poore pipeline ki raksha karta hai pehle $g$ fix karke.

Equipment checklist

Khud test karo — right side cover karo.

Ek curve pe

y^{'}

kya measure karta hai?

Har point pe slope (steepness);

y^{''}

hai kitni tez woh slope change hoti hai (curve kitni sharply bend karti hai).

y^{''} + p y^{'} + q y = g

mein "push" kaun sa piece hai?

g (x)

, right-hand side pe forcing.

Equation homogeneous kab hoti hai?

Jab uska right-hand side

0

ho (koi external push nahi).

Hume exactly do solutions

y_{1}, y_{2}

kyun chahiye?

Second-order equation ke do independent free choices hote hain, toh uski free motion do building blocks ka blend hai.

Variation of parameters define karne wala ek swap kya hai?

Constant strengths

c_{1}, c_{2}

ko varying functions

u_{1} (x), u_{2} (x)

se replace karo.

2 \times 2

determinant

a c b d

compute karo.

a d - b c

Determinant bars aur absolute-value bars mein farq kaise pehchanoge?

Absolute value ek single number ko wrap karta hai (

∣ - 3 ∣ = 3

, kabhi negative nahi); determinant chaar numbers ke grid ko wrap karta hai (

a d - b c

, negative ho sakta hai).

y_{1}, y_{2}

ka Wronskian likho.

W = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}

W = 0

kya warn karta hai, aur ek baar check kyun kaafi hai?

Do solutions independent nahi hain aur method fail ho jaata hai; Abel's identity

W = W (x_{0}) e^{- \int p d x}

dikhata hai ki

W

ya toh interval pe har jagah zero hai ya kaheen nahi.

Cramer's rule se

u_{1}^{'}

aur

u_{2}^{'}

kya aate hain?

u_{1}^{'} = - y_{2} g / W

aur

u_{2}^{'} = + y_{1} g / W

Cramer's rule ke baad integrate kyun karte hain, aur

+ C

kyun drop karte hain?

Integrate karna prime undo karta hai

u_{1}, u_{2}

recover karne ke liye; constants sirf

C_{1} y_{1} + C_{2} y_{2}

add karti, jo ek homogeneous piece hai jo already

c_{1} y_{1} + c_{2} y_{2}

mein count hai.

y_{p}

ka final formula likho.

y_{p} = - y_{1} \int (y_{2} g / W) d x + y_{2} \int (y_{1} g / W) d x

g

read karne se pehle kya check karna hai?

ODE ko standard form (leading coefficient

1

) mein daalo, divide karke.

Connections

Wronskian — Section 7 mein banaya fairness scale $W$ .
Cramer's Rule — Section 8 mein do-equation system solve karne ka shortcut.
Second-order linear homogeneous ODE — jahan se building blocks $y_{1}, y_{2}$ aate hain.
Reduction of Order — $y_{2}$ nikalne ka tarika jab sirf $y_{1}$ pata ho.
Method of Undetermined Coefficients — special $g (x)$ ke liye faster, narrower cousin.
Green's function — isi idea ko integral kernel ki tarah recast karta hai.