Case 2 - repeated real root — reduction of order: Foundations

4.6.12 · D1 · Maths › Ordinary Differential Equations › Case 2 - repeated real root — reduction of order

Yeh page ground floor hai. Yeh assume karta hai ki tumne kabhi derivative, exponential, ya differential equation nahi dekha. Parent note ka har symbol yahan ek ek brick karke build kiya gaya hai.

1. Function kya hoti hai, aur uska graph kaisa dikhta hai

Ise ek curve ki tarah picture karo jo ek grid par khichi gayi ho. Horizontal axis input $x$ hai; vertical axis output $y$ hai. Curve par har point ek input-output pair hai.

Figure s01 — ek function ek curve ke roop mein. Har vertical dashed line ek input $x$ ko uske single output $y$ se pin karti hai.

Figure — Case 2 - repeated real root — reduction of order

Topic ko yeh kyun chahiye: poora chapter ek unknown curve $y (x)$ dhundhne ke baare mein hai. Baaki sab cheezein — slopes, characteristic equations — yeh machinery hai jo yeh pin karne ke liye hai ki woh curve kaunsi hai.

2. Derivative $y^{'}$ — slope, limit se define ki gayi

Hume "ek chhota step" ko exact banana hoga. $x$ se $x + h$ tak size $h$ ka ek step lo. Do curve-points $(x, y (x))$ aur $(x + h, y (x + h))$ ko joinе karti line — ek secant line — ka slope hai $\frac{y ( x + h ) - y ( x )}{h} (rise over run) .$

Figure s02 — secant $h \to 0$ ke saath tangent ban jaati hai. Blue secants pink tangent ki taraf pivot karti hain; rise-over-run triangle exactly dikhata hai ki $y^{'}$ kya measure karta hai.

Yeh tool kyun, koi aur kyun nahi? Hume ek tarika chahiye kitni tez koi quantity change hoti hai — velocity, growth, decay — ke baare mein baat karne ka. Derivative precisely woh machine hai jo "right here kitna steep?" ka jawab deti hai. Figure s02 mein pink tangent line dekho: uski steepness hi $y^{'}$ ki value hai.

3. Second derivative $y^{''}$ — curvature, slope-of-the-slope

Figure s03 — curvature ka sign hota hai. Blue "valley" upar bend karti hai ( $y^{''} > 0$ ), pink "hill" neeche bend karti hai ( $y^{''} < 0$ ), yellow straight line ka $y^{''} = 0$ hai.

Topic ko yeh kyun chahiye: equation ko second-order isliye kaha jaata hai kyunki $y^{''}$ usme appear karta hai. Physically $y^{''}$ acceleration hai; ek spring, ek circuit, ek swinging pendulum sab position ( $y$ ), velocity ( $y^{'}$ ) aur acceleration ( $y^{''}$ ) ko relate karne wale rules follow karte hain. Woh relation hi hamaari differential equation hai.

Recall Primes par quick self-check

Agar $y^{'}$ slope hai aur $y^{''}$ slope-of-the-slope hai, toh $y^{''} = 0$ geometrically kya matlab rakhta hai? ::: Slope kabhi nahi badlta — curve ek straight line hai.

4. Chain rule — function of a function ko differentiate karna

Topic ko yeh kyun chahiye: hum constantly $e^{r x}$ , $x e^{r x}$ , etc., differentiate karte hain jahan andar ek rate $r$ baitha hota hai. Chain rule woh tool hai jo us inner rate $r$ ko aage nikalta hai — §5 mein turant use hoga.

5. Factorials aur exponential $e^{r x}$ — woh function jo apna khud ka slope hai

Ab hum $e^{x}$ ko kuch nahi se build karte hain aur §2 power rule aur §4 chain rule se uski slope rule derive karte hain, kyunki poora topic isi par tikaa hai. Pehle, ek notation ka tukda.

Recall "Sum ko term by term differentiate karne" par ek baat

"Differentiate" aur "infinitely many terms add karo" ke order ko swap karna har infinite sum ke liye automatic nahi hota. Yeh yahan legal hai kyunki yeh particular series converges hoti hai (iske terms itni tez shrink hote hain — factorial denominators kisi bhi fixed $x$ se aage nikal jaate hain — ki tail essentially kuch contribute nahi karta), aur aisi tezi se converge hone wali power series ke liye calculus ka ek theorem guarantee karta hai ki term-by-term derivative valid hai. Hum woh theorem as given lete hain; payoff yeh clean rule hai $\frac{d}{d x} e^{x} = e^{x}$ . ::: Term-by-term differentiation isliye justified hai kyunki exponential series itni tez converge hoti hai (factorials $x^{n}$ se aage nikal jaate hain).

Figure s04 — $r$ ke teen signs ke liye exponential. Pink grow karta hai ( $r > 0$ ), blue decay karta hai ( $r < 0$ ), yellow flat line $e^{0} = 1$ hai.

Agar $r > 0$ : $e^{r x}$ grow karta hai (rightward explode hota hai).
Agar $r < 0$ : $e^{r x}$ decay karke $0$ ki taraf jaata hai (settle ho jaata hai) — pale-blue curve.
Agar $r = 0$ : $e^{0} = 1$ , ek flat constant line.
Note karo $e^{r x} > 0$ hamesha — yeh kabhi zero ko hit ya cross nahi karta. Isliye hum baad mein safely " $e^{r x}$ se divide" kar sakte hain.

Intuition Hum

$e^{r x}$ trial kyun karte hain naa ki $sin$ ya $x^{2}$ Hume ek aisi function chahiye jiske derivatives bas khud ke multiples hon, taaki equation (number) $\times e^{r x}$ mein collapse ho jaaye. Exponential yeh kaam ek single multiplier $r$ se har baar karta hai. Periodic functions jaise $sin x$ aur $cos x$ bhi differentiation ke under cycle karte hain ( $sin \to cos \to - sin \to \dots$ ), lekin woh ek doosre ki jagah lete hain instead of same function ka clean multiple return karne ke — toh ek single $sin$ equation ko khud se collapse nahi karega. (Remarkably, $sin$ aur $cos$ secretly complex $r$ ke through exponentials disguise mein hain — woh reunion exactly Case 3 complex roots hai.) $x^{2}$ jaise powers differentiation ke under puri tarah shape badal lete hain ( $x^{2} \to 2 x \to 2$ ), toh woh kabhi khud ko reproduce nahi karte. Yahi $e^{r x}$ ko natural trial chhodhta hai.

6. Characteristic equation $a r^{2} + b r + c = 0$ aur uska discriminant

Jab discriminant $0$ hota hai, $\pm$ vanish ho jaata hai aur dono roots single value mein merge ho jaate hain $r = - \frac{b}{2 a} .$ Woh collapse hi Case 2 ka poora drama hai: do roots ek ban gaye, toh hum ek solution short hain. ODE se quadratic tak is transfer ki full derivation ke liye Characteristic equation of linear ODEs dekho.

7. Dimension, linear independence, aur Wronskian

8. Equipment checklist

Khud test karo — tum parent topic ke liye ready ho jab har ek ka jawab de sako:

y^{'}

ka geometrically kya matlab hai?

Curve

y (x)

ka har point par slope (steepness).

y^{'}

ki formal limit definition kya hai?

y^{'} (x) = lim_{h \to 0} \frac{y ( x + h ) - y ( x )}{h}

— secant slope ki settling-value jab step

h

zero ki taraf shrink hoti hai.

Limit

h \to 0

kya hota hai?

Woh single number jise ek expression approach karta hai jab

h

0

ki taraf shrink hota hai bina kabhi

0

ke barabar hue.

Power rule kya hai aur yeh derive kaise hota hai?

\frac{d}{d x} x^{n} = n x^{n - 1}

;

\frac{( x + h ) ^{n} - x ^{n}}{h}

expand karke aur

h \to 0

let karke derive hota hai.

Chain rule kya kehta hai?

\frac{d}{d x} f (u (x)) = f^{'} (u) \cdot u^{'} (x)

— outside ka slope times inside ka slope.

n!

kya hai?

Factorial:

n \cdot (n - 1) \dots 2 \cdot 1

0! = 1

ke saath; jaise

4! = 24

y^{''}

ka geometrically kya matlab hai?

Slope-of-the-slope — curve kitna bend karta hai (uski curvature).

e^{x}

ko scratch se kaise define kiya jaata hai?

Power series

1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots

ke roop mein;

x = 1

set karne par

e \approx 2.718

milta hai.

e^{x}

apni khud ki derivative kyun hai?

Term-by-term power-rule differentiation har

\frac{x ^{n}}{n !}

\frac{x ^{n - 1}}{( n - 1 )!}

par bhejti hai, wahi sum reproduce karte hue.

e^{r x}

ki derivative kya hai, aur kyun?

r e^{r x}

— chain rule self-reproducing outside ko inside ke slope

r

se multiply karta hai.

Is page par

r

kis tarah ka number hai?

Ek fixed real constant (positive, negative, ya zero); complex case Case 3 par defer kiya gaya hai.

Hum

e^{r x}

trial kyun karte hain naa ki

sin x

x^{2}

Sirf

e^{r x}

differentiation ke under khud ka ek clean multiple return karta hai;

sin / cos

ek doosre mein swap hote hain, aur powers shape change kar lete hain.

Hum equation ko

e^{r x}

se hamesha divide kyun kar sakte hain?

Kyunki

e^{r x} > 0

har

x

ke liye — yeh kabhi zero nahi hota.

ODE se characteristic equation kaise milti hai?

y = e^{r x}

substitute karo,

e^{r x}

factor out karo,

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

bachta hai.

Discriminant

b^{2} - 4 a c

kya batata hai?

Uska sign case pick karta hai: positive = distinct roots, zero = repeated root, negative = complex roots.

Jab

b^{2} - 4 a c = 0

toh single repeated root kya hota hai?

r = - \frac{b}{2 a}

"2-dimensional solution space" ka kya matlab hai?

Exactly do independent solution-curves chahiye taaki har solution

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

ka combination ho.

Wronskian

W \neq = 0

kya prove karta hai, aur kyun?

Linear independence — kyunki ek scaled copy

y_{2} = k y_{1}

W = 0

force karta hai, toh

W \neq = 0

uss possibility ko rule out karta hai.

a, b, c

aur

C_{1}, C_{2}

mein kya fark hai?

a, b, c

ODE ke fixed coefficients hain;

C_{1}, C_{2}

general solution mein free constants hain jo initial conditions meet karne ke liye choose kiye jaate hain.

Foundations — Case 2 - repeated real root — reduction of order

1. Function kya hoti hai, aur uska graph kaisa dikhta hai

2. Derivative $y^{'}$ — slope, limit se define ki gayi

3. Second derivative $y^{''}$ — curvature, slope-of-the-slope

4. Chain rule — function of a function ko differentiate karna

5. Factorials aur exponential $e^{r x}$ — woh function jo apna khud ka slope hai

6. Characteristic equation $a r^{2} + b r + c = 0$ aur uska discriminant

7. Dimension, linear independence, aur Wronskian

8. Equipment checklist

Prerequisite map