Homogeneous with constant coefficients — characteristic equation

4.6.10 · Maths › Ordinary Differential Equations

1. Setup aur definitions

KYUN homogeneous + linear hone se hum solutions add kar sakte hain? Kyunki operator $L [y] = a_{n} y^{(n)} + \dots + a_{0} y$ linear hai: $L [c_{1} y_{1} + c_{2} y_{2}] = c_{1} L [y_{1}] + c_{2} L [y_{2}]$ . Agar $L [y_{1}] = 0$ aur $L [y_{2}] = 0$ , toh koi bhi combination bhi $0$ hoga. Toh general solution ek linear combination of independent basic solutions hai — aur $n$ -th order equation ko $== n == "$ independent ones chahiye.

2. Scratch se derivation (second order)

Lo $a y^{''} + b y^{'} + cy = 0$ , $a \neq = 0$ .

Step 1 — Guess karo $y = e^{r x}$ . Yeh step kyun? Sirf exponentials hi differentiation ke baad khud ko reproduce karte hain, isliye yeh natural candidate hain combination cancel karne ke liye.

Step 2 — Differentiate karo: $y^{'} = r e^{r x}$ , $y^{''} = r^{2} e^{r x}$ .

Step 3 — Substitute karo: $a r^{2} e^{r x} + b r e^{r x} + c e^{r x} = 0 \Rightarrow e^{r x} (a r^{2} + b r + c) = 0.$ Yeh step kyun? Common $e^{r x}$ factor out karo taaki algebra isolate ho sake.

Step 4 — $e^{r x}$ se divide karo (kabhi zero nahi hota): $a r^{2} + b r + c = 0$ yahi characteristic equation hai. Quadratic formula se solve karo $r = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .

Discriminant $Δ = b^{2} - 4 a c$ sab kuch 3 cases mein split kar deta hai.

3. Teen cases (aur KYUN har form aati hai)

Repeated root ko $x e^{r x}$ kyun chahiye?

Agar $r$ double root hai toh $a r^{2} + b r + c = a (r - r_{0})^{2}$ , toh operator factor ho jaata hai $a (D - r_{0})^{2} y = 0$ jahaan $D = \frac{d}{d x}$ . $(D - r_{0}) u = 0$ solve karne par milta hai $u = c_{2} e^{r_{0} x}$ , phir $(D - r_{0}) y = u$ ek first-order linear ODE hai jiska solution hai $y = (c_{1} + c_{2} x) e^{r_{0} x}$ . Factor $x$ isliye aata hai kyunki integrating factor exponential cancel kar deta hai aur integrate karne ke liye ek constant bachta hai, jo $x$ mein linear term produce karta hai. Issi wajah se — yeh koi magic rule nahi hai.

Complex roots sin aur cos kyun dete hain?

$r = α \pm i β$ ke saath, solutions hain $e^{(α + i β) x}$ aur $e^{(α - i β) x}$ . Yeh complex hain; lekin ODE mein real coefficients hain, toh humein real solutions chahiye. Real aur imaginary parts lo: $e^{α x} cos β x$ aur $e^{α x} sin β x$ — dono solutions hain aur independent hain. (Yeh sab kaam Euler's formula kar raha hai.)

Figure — Homogeneous with constant coefficients — characteristic equation

4. Worked examples

5. Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek jhula socho. Yeh rule jo batata hai ki jhula kaise move karta hai, sirf jhule ki position, uski speed, aur uski speed kitni tezi se badh rahi hai — inhe fixed numbers se multiply karke use karta hai. Hum bet lagate hain ki answer ek khaas number $e$ ko power par raise karne jaisa dikhega, kyunki yahi ek aisi shape hai jo waise hi rehti hai jab tum maapte ho ki yeh kitni tezi se badal rahi hai. Bet daalne par yeh mushkil "rate of change" ka sawaal ek aasaan "number puzzle solve karo" ( $a r^{2} + b r + c = 0$ ) mein badal jaata hai. Agar puzzle do normal numbers de → do stretchy exponential curves. Agar ek repeated number de → tum ek extra $x$ ghussa dete ho. Agar "imaginary" numbers de → jhula actually wiggle karta hai, toh waves milti hain (sin aur cos), shayad fade hoti hui.

Connections

Linear Differential Operators and Superposition
Euler's Formula and Complex Exponentials
Wronskian and Linear Independence of Solutions
Method of Undetermined Coefficients (non-homogeneous next step)
Damped Harmonic Oscillator (physics application of Case 3)
Second Order Linear ODE — Reduction of Order (explains $x e^{r x}$ )

Flashcards

Constant-coefficient linear ODEs ke liye

y = e^{r x}

kyun guess karte hain?

Exponentials hi aisi akeli functions hain jinke derivatives khud ki scaled copies hote hain, isliye yeh ODE ko

r

mein ek polynomial equation mein badal deti hain.

a y^{''} + b y^{'} + cy = 0

ki characteristic equation kya hai?

a r^{2} + b r + c = 0

, jo

y = e^{r x}

substitute karke aur

e^{r x}

se divide karke milti hai.

Case

Δ > 0

(distinct real roots) ka general solution?

y = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}

Case

Δ = 0

(repeated root

r

) ka general solution?

y = (c_{1} + c_{2} x) e^{r x}

Case

Δ < 0

(roots

α \pm i β

) ka general solution?

y = e^{α x} (c_{1} cos β x + c_{2} sin β x)

Repeated root ko

x

ka extra factor kyun chahiye?

Operator

(D - r_{0})^{2}

mein factor hota hai; chained first-order equations solve karne par ek linear-in-

x

term aata hai, jo doosra independent solution

x e^{r x}

deta hai.

Complex roots sin aur cos kyun produce karte hain?

Euler's formula:

e^{(α \pm i β) x} = e^{α x} (cos β x \pm i sin β x)

; real aur imaginary parts lene par do real independent solutions milte hain.

r = α \pm i β

mein

α

β

kya control karta hai?

α

envelope ki exponential growth/decay control karta hai;

β

oscillation frequency control karta hai.

y^{''} + 4 y = 0

solve karo.

r^{2} + 4 = 0 \Rightarrow r = \pm 2 i

, toh

y = c_{1} cos 2 x + c_{2} sin 2 x

Order-

n

homogeneous linear ODE mein kitne independent solutions hote hain?

Exactly

n