Homogeneous with constant coefficients — characteristic equation: Question bank

4.6.10 · D5 · Maths › Ordinary Differential Equations › Homogeneous with constant coefficients — characteristic equa

Quick symbol reminder taaki yahan koi cheez unexplained na ho:

$y^{'} = \frac{d y}{d x}$ ka matlab hai "jab $x$ move karta hai toh $y$ kitni tez change hoti hai"; $y^{''}$ yeh hai ki woh rate kitni tez change hoti hai.
$r$ woh number hai jo guess $y = e^{r x}$ mein hai — characteristic equation $a r^{2} + b r + c = 0$ ki root.
$Δ = b^{2} - 4 a c$ discriminant hai: square root ke andar woh cheez jo decide karti hai ki aap teen cases (Different / Double / Dizzy) mein se kisMein ho.
$α \pm i β$ ka matlab hai ek complex root jisme real part $α$ hai (growth/decay) aur imaginary part $β$ hai (wiggle speed).

True or false — justify karo

TF1. "Har second-order homogeneous constant-coefficient ODE mein exactly do independent solutions hoti hain."

True — order 2 hai, aur operator ki linearity ek 2-dimensional solution space guarantee karti hai; ek repeated root bhi do independent pieces

e^{r x}

aur

x e^{r x}

deti hai.

TF2. "Agar

y_{1}

aur

y_{2}

dono

L [y] = 0

solve karte hain, toh

3 y_{1} - 7 y_{2}

bhi ise solve karta hai."

True — operator

L

linear hai, isliye

L [3 y_{1} - 7 y_{2}] = 3 L [y_{1}] - 7 L [y_{2}] = 0

; yeh superposition sirf isliye kaam karti hai kyunki equation homogeneous hai (right side

0

hai). Dekho Linear Differential Operators and Superposition.

TF3. "Substituted equation ko

e^{r x}

se hamesha divide kiya ja sakta hai taaki polynomial mile."

True —

e^{r x}

kisi bhi real ya complex

x

ke liye kabhi zero nahi hota, isliye divide karna legal hai aur koi solution nahi khota; yeh woh step hai jo calculus ko algebra mein badal deta hai.

TF4. "

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

ke liye characteristic equation

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

hai."

False — yeh

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

hai; leading

a

drop karne se dono roots aur discriminant

Δ = b^{2} - 4 a c

change ho jaate hain, jab tak

a = 1

na ho.

TF5. "Complex roots ka matlab hai ODE ke koi real solutions nahi hain."

False — ODE ke paas phir bhi purely real solutions

e^{α x} cos β x

aur

e^{α x} sin β x

hote hain; complex roots sirf ek intermediate bookkeeping device hain, jo Euler ke through real form mein convert hote hain. Dekho Euler's Formula and Complex Exponentials.

TF6. "Ek repeated root hamesha

Δ = 0

correspond karta hai."

True — double root tab exist karta hai jab quadratic formula mein square root vanish ho jaye, yaani

b^{2} - 4 a c = 0

, jo single value

r = - b / (2 a)

deta hai.

TF7. "Agar

α < 0

toh solution decay karta hai; agar

α > 0

toh blow up karta hai."

True — envelope

e^{α x}

hai, jo

α < 0

hone par shrink karta hai aur

α > 0

hone par grow karta hai; yeh exactly ek Damped Harmonic Oscillator ka growth/decay knob hai.

TF8. "

e^{2 x}

aur

e^{3 x}

dependent ho sakte hain agar constants theek se line up karen."

False — distinct-exponent exponentials hamesha linearly independent hote hain; unka Wronskian hai

(3 - 2) e^{5 x} \neq = 0

sabhi

x

ke liye. Dekho Wronskian and Linear Independence of Solutions.

TF9. "

β

(imaginary part) ki value set karti hai ki solution kitni tez oscillate karta hai."

True —

β

angular frequency hai

cos β x

aur

sin β x

ke andar; bada

β

matlab har unit

x

mein zyada wiggles, jabki

α

frequency ko untouched chodta hai.

Error dhundo

SE1. "Repeated root

r = 3

, toh

y = c_{1} e^{3 x} + c_{2} e^{3 x}

Dono terms

(c_{1} + c_{2}) e^{3 x}

mein collapse ho jaate hain — ek single constant, isliye yeh sirf ek solution hai; sahi second solution

x e^{3 x}

hai, jo

y = (c_{1} + c_{2} x) e^{3 x}

deta hai.

SE2. "Roots

r = - 1 \pm 2 i

, toh answer hai

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

aur hum ruk jaate hain."

Galat nahi, lekin real problem ke liye incomplete hai: Euler se real form mein convert karo

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

taaki solution real-valued ho.

SE3. "

y^{''} + 4 y = 0

ke liye, char. eq.

r^{2} + 4 = 0

r = \pm 4

milta hai."

Error:

r^{2} = - 4

hai,

16

nahi, isliye

r = \pm 2 i

hai,

\pm 4

nahi; roots imaginary hain, jo oscillation

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

dete hain.

SE4. "

2 y^{''} + 3 y^{'} + y = 0

ke liye discriminant hai

Δ = 3^{2} - 4 (1) (1) = 5

Error:

a = 2

c = 1

, isliye

Δ = b^{2} - 4 a c = 9 - 8 = 1

hai,

5

nahi; leading coefficient

a

bhool jaana classic slip hai.

SE5. "

y = e^{- x} sin 2 x

; differentiate karne par

y^{'} = e^{- x} \cdot 2 cos 2 x

milta hai."

Error: yeh ek product hai, isliye product rule zaroori hai —

y^{'} = - e^{- x} sin 2 x + 2 e^{- x} cos 2 x

; pehli term drop karna decay ke contribution ko kho deta hai.

SE6. "Kyunki

r = 0

root hai

r^{2} - 3 r = 0

ki, solution

e^{0 \cdot x}

trivial hai aur ignore kiya ja sakta hai."

Error:

e^{0 x} = 1

ek genuine, non-trivial constant solution hai; poora answer hai

y = c_{1} + c_{2} e^{3 x}

, aur constant drop karna poori ek dimension kho deta hai.

SE7. "

y^{'''} - y = 0

ki characteristic equation

r^{2} - 1 = 0

hai."

Error: order 3 hai, isliye yeh

r^{3} - 1 = 0

hona chahiye; polynomial degree hamesha ODE order ke barabar hoti hai.

Why questions

WHY1. Hum polynomial ya sine ki jagah exponential guess kyun karte hain?

Kyunki differentiation ko usi function ki scaled copy return karni hogi taaki linear combination zero mein cancel ho sake, aur

e^{r x}

woh unique family hai jisme

\frac{d}{d x} e^{r x} = r e^{r x}

; sines/cosines sirf tab aate hain jab

r

complex hota hai, jaise exponentials ki disguise.

WHY2. Double root par extra factor

x

kyun aata hai,

x^{2}

ln x

kyun nahi?

Operator ko

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

factor karke aur resulting first-order chain solve karne par exactly ek constant integrate hoti hai, jo linear term

c_{2} x

produce karta hai; triple root ke liye phir

x^{2}

chahiye hoga. Dekho Second Order Linear ODE — Reduction of Order.

WHY3. Superposition kaam karne ke liye ODE homogeneous kyun honi chahiye?

Superposition ke liye chahiye

L [c_{1} y_{1} + c_{2} y_{2}] = c_{1} L [y_{1}] + c_{2} L [y_{2}] = 0

, jiske liye dono pieces ko

0

hit karna zaroori hai; nonzero right side ke saath combination forcing terms sum karta, cancel nahi karta. Dekho Method of Undetermined Coefficients.

WHY4. Complex root ka real part

α

oscillation frequency ko affect kyun nahi karta?

Kyunki

α

envelope

e^{α x}

mein rehta hai (pure amplitude scaling) jabki frequency

cos β x

ke andar

β

mein poori tarah rehti hai; ek wave ko shrinking amplitude se multiply karna uski size change karta hai, kitni baar zero cross karta hai yeh nahi.

WHY5. Hum complex solution ke real aur imaginary parts le kar do independent real solutions kyun pa sakte hain?

Kyunki ODE ke real coefficients hain, isliye agar

u + i v

ise solve karta hai toh

u

aur

v

alag-alag solve karte hain;

cos β x

aur

sin β x

ek doosre ke scalar multiples nahi hain, isliye woh independent hain (nonzero Wronskian).

WHY6. Discriminant

Δ

, individual coefficients nahi, solution ki shape kyun decide karta hai?

Δ = b^{2} - 4 a c

exactly woh quantity hai jo

r = \frac{- b \pm Δ}{2 a}

mein square root ke andar hai, isliye sirf iski sign decide karti hai ki roots do reals hain, ek real hai, ya complex pair hai — yahi case fix karta hai.

WHY7.

e^{r x}

se divide karna,

y

se divide karne se safer kyun hai?

e^{r x}

provably nonzero hai har jagah, isliye koi solution nahi khota;

y

se divide karna

y = 0

wali jagah destroy kar sakta hai aur spurious restrictions introduce kar sakta hai.

Edge cases

EC1. "Agar ek root

r = 0

ho toh?"

Toh

e^{0 x} = 1

ek constant solution hai;

r = 0

aur

r = k

ke liye answer hai

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

, aur

0

par double root

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

deta hai.

EC2. "Agar dono roots

r = 0

hon (double root at zero)?"

Equation essentially

y^{''} = 0

hai, isliye

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

—

r = 0

ke saath

x e^{r x}

rule ek straight line mein collapse ho jata hai, jo plain double integration se match karta hai.

EC3. "Agar roots

α = 0

ke saath complex hon, jaise

\pm i β

Envelope

e^{0 x} = 1

disappear ho jata hai, sirf undamped pure oscillation

y = c_{1} cos β x + c_{2} sin β x

bachti hai — ideal frictionless spring.

EC4. "Agar

a = 0

ho, toh

y^{''}

term nahi hoga?"

Phir yeh second-order hai hi nahi; yeh first-order equation

b y^{'} + cy = 0

mein drop ho jata hai single root

r = - c / b

ke saath aur solution

y = c_{1} e^{r x}

— poora three-case machinery apply nahi hota.

EC5. "Agar

b = 0

aur

c > 0

ho, jaise

y^{''} + cy = 0

Phir automatically

Δ = - 4 a c < 0

hoga, pure imaginary roots aur undamped oscillation milegi;

b

damping term hai, isliye ise remove karna saari decay remove kar deta hai.

EC6. "Agar do exponents bahut close hain lekin distinct hain, jaise

r = 3

aur

r = 3.0001

Woh Case 1 hi rahenge do alag exponentials ke saath; jaise gap zero ki taraf shrink hota hai, pair

e^{r_{1} x}, e^{r_{2} x}

continuously

e^{r x}, x e^{r x}

mein deform ho jaata hai, yahi wajah hai ki repeated case

x

factor borrow karta hai — yeh distinct case ki limit hai.

EC7. "Agar initial conditions

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

par

x = 0

pe apply ki jaayein?"

x = 0

par envelope

e^{0} = 1

hai,

cos 0 = 1

sin 0 = 0

, isliye

y (0) = c_{1}

immediately milta hai; lekin

y^{'} (0)

ke liye substitute karne se pehle envelope aur trig factor dono par full product rule zaroori hai.

Connections

Linear Differential Operators and Superposition
Euler's Formula and Complex Exponentials
Wronskian and Linear Independence of Solutions
Method of Undetermined Coefficients
Damped Harmonic Oscillator
Second Order Linear ODE — Reduction of Order