Cauchy-Euler (Equidimensional) equation: Foundations

4.6.16 · D1 · Maths › Ordinary Differential Equations › Cauchy-Euler (Equidimensional) equation

Parent note ki ek bhi line padhne se pehle, aapko usmein har ek squiggle pehchanna hoga. Yeh page unhe ek-ek karke banata hai — pehle plain words mein, phir picture, phir kyun yeh topic uske bina nahi chal sakta. (Ek baat: ODE mein "ordinary" ka matlab sirf yeh hai ki sirf ek input variable $x$ hai; "differential" ka matlab hai ki derivatives aate hain.)

1. Variable $x$ aur function $y$

Picture: ek horizontal $x$ -axis ke upar ek curve draw karo. Koi bhi $x$ chuno; seedha upar curve tak jao; woh height hi $y$ hai.

Topic ko yeh kyun chahiye: ek ODE (ordinary differential equation) ek aisa puzzle hai jiska jawab ek poora curve $y (x)$ hota hai, koi single number nahi. Baaki sab machinery hai uss curve ko dhundhne ke liye.

2. Derivative $y^{'}$ aur uske cousins $y^{''}$ , $y^{(k)}$

Picture: curve par kisi point par tangent line draw karo. Uski steepness hi $y^{'}$ hai. Jahaan curve smile ki tarah upar bend karta hai, $y^{''} > 0$ ; frown ki tarah neeche, $y^{''} < 0$ .

Ek fact jis par hum zyada bharosa karenge: ek power ko differentiate karne se uska exponent ek se ghatta hai. $\frac{d}{d x} x^{m} = m x^{m - 1}, \frac{d ^{2}}{d x ^{2}} x^{m} = m (m - 1) x^{m - 2} .$ " $m x^{m - 1}$ " zor se bolo: exponent ko neeche saamne laao, usme se ek ghatao. Bas itna hi differentiation kisi power ke saath karta hai. Zyada generally $\frac{d ^{k}}{d x ^{k}} x^{m} = m (m - 1) \dots (m - k + 1) x^{m - k}$ — har ek $k$ derivative exponent ko ek se ghatata hai, toh $k$ steps ke baad power $x^{m - k}$ hai.

3. Powers $x^{m}$ — poore, negative, aur fractional exponents

Picture: ek axis par $x^{2}$ (upar kholti hui bowl), $x^{1}$ (seedha ramp), aur $x^{- 1} = 1/ x$ (axis ki taraf girta curve) draw karo. Same family " $x$ to a power", $m$ ke hisaab se bilkul alag shapes.

4. Natural log $ln x$ — aur $x^{m} ln x$ kahaan se aata hai

Picture: ek curve jo $(1, 0)$ se guzarta hai, hamesha badhta hai par zyada se zyada dheere, aur $x \to 0^{+}$ par $- \infty$ mein ghus jaata hai. $y$ -axis ke baayein koi values nahi hain — isliye topic baar baar warn karta hai " $x > 0$ ".

5. Exponential $e^{t}$ aur imaginary $i$

Picture: $θ$ ko ek angle samjho; $e^{i θ}$ ek point hai jo unit circle par ghoom raha hai. Poori geometry ke liye Euler's Formula dekho.

Topic ko yeh kyun chahiye: jab quadratic ka answer $i$ contain karta hai (complex roots $α \pm i β$ ), toh power $x^{α \pm i β}$ ek imaginary exponent carry karta hai. Euler's formula use real $cos (β ln x)$ aur $sin (β ln x)$ mein badal deta hai — ek aisa solution jo tum actually plot kar sako.

6. Substitution $t = ln x$ aur chain rule

7. Quadratic $a m^{2} + bm + c = 0$ aur uske roots

Topic ko yeh kyun chahiye: $y = x^{m}$ guess karne ke baad, poora ODE $m$ mein ek quadratic mein collapse ho jaata hai — indicial (auxiliary) equation, Characteristic / Auxiliary Equation ka cousin. Teen root-types mein se kaun sa milta hai yeh answer ki shape decide karta hai (distinct powers / power-with- $ln$ / spinning cosines).

8. ODE ki order, aur constants $C_{1}, C_{2}$

Picture: curves ki ek poori fan, har ek $(C_{1}, C_{2})$ choice ke liye ek; initial conditions baad mein pin karte hain ki tum konsa ek curve chahte ho.

Yeh sab topic ko kaise feed karta hai

Neeche wala diagram ek dependency map hai: har box is page ka ek foundation hai, aur ek arrow "A → B" ka matlab hai "B samajhne se pehle A chahiye." Arrows ko neeche follow karo aur tum literally poore topic ki logic ko re-trace kar rahe ho — top par raw symbols se lekar bottom par finished Cauchy-Euler solution tak.

Kaise padhen: function aur uske derivatives (top left) plus power family guess $y = x^{m}$ produce karte hain; guess plus roots indicial quadratic produce karte hain; quadratic ke teen root-types, $ln x$ , Euler's formula, aur $t = ln x$ substitution ke saath milkar teen solution cases assemble karte hain; finally constants $C_{1}, C_{2}$ unhe blend karke general Cauchy-Euler solution banate hain.

Disguise-removal (rule 6) Reduction of Order se connect hota hai (ek solution se doosra solution dhundhna) aur, un equations ke liye jo equidimensional nahi hain, Frobenius Method & Regular Singular Points se. Non-homogeneous versions Variation of Parameters use karte hain.

Equipment checklist

Khud test karo — right side cover karo.

ODE abbreviation ka matlab kya hai?

Ordinary differential equation — ek input variable, derivatives aate hain.

y^{(k)}

ka kya matlab hai, aur

y^{(0)}

kya hai?

y

ko total

k

baar differentiate karo;

y^{(0)} = y

(kuch mat karo).

y^{'}

curve par kya measure karta hai?

Us point par

y

ki slope (steepness).

y^{''}

kya measure karta hai?

Slope khud kaise change hoti hai — bending / curvature.

x^{m}

ko ek baar differentiate karo.

m x^{m - 1}

— exponent ko neeche laao, ek ghatao.

x^{m}

ko do baar differentiate karo.

m (m - 1) x^{m - 2}

x^{- 1}

kya hai, aur

x^{1/2}

kya hai?

1/ x

; aur

x

Jab

m

fractional/irrational ho toh

x^{m}

safely kis

x

ke liye defined hai, aur kyun?

Sirf

x > 0

; negatives ke roots aur branch choices

x \leq 0

ke liye toot jaate hain.

x^{0}

kya hai?

1

ln x

kaunsa sawaal answer karta hai, aur kis

x

ke liye defined hai?

e

ko kis power par uthao ki

x

mile?"; sirf

x > 0

ke liye defined hai.

x^{k} \frac{d ^{k}}{d x ^{k}} x^{m}

ek constant times

x^{m}

kyun deta hai?

Differentiate karne ka

- k

aur coefficient ka

+ k

cancel ho jaate hain:

x^{k} \cdot x^{m - k} = x^{m}

Euler's formula batao.

e^{i θ} = cos θ + i sin θ

i

kya hai?

Imaginary unit jisme

i^{2} = - 1

Kaun sa substitution Cauchy-Euler ko un-disguise karta hai, aur uski key identity kya hai?

t = ln x

; phir

x \frac{d y}{d x} = \frac{d y}{d t}

Repeated root ke liye doosra solution

x^{m} ln x

kahaan se aata hai?

Reduction of order:

v x^{m}

try karo; factor

v

integrate hokar

ln x

banta hai.

ODE ki order kya hoti hai, aur kitne arbitrary constants carry karta hai?

Sabse bada derivative jo aata ho; ek

n

-th-order ODE

n

constants carry karta hai.

Indicial quadratic ke kitne roots hote hain, aur teen types kaunse hain?

Do roots: distinct real, repeated, ya complex.