Second-order linear ODEs — superposition principle, general theory

4.6.9 · Maths › Ordinary Differential Equations

Second-order linear ODE kya hota hai?

YEH FORM KYUN? $a y^{''} + b y^{'} + cy = h$ mein leading coefficient $a (x)$ se divide karne par upar wali "monic" form milti hai. Hum yeh isliye karte hain taaki theory (existence, Wronskian) saaf padhti rahe.

Linear Operator viewpoint (har cheez ka engine)

Superposition Principle

General solution structure

Linear independence & Wronskian

Worked Example 1 — general solution banana aur verify karna

Worked Example 2 — non-homogeneous structure

Existence & Uniqueness (is theory ko use karne ka licence)

Common mistakes (steel-manned)

Flashcards

2nd-order linear ODE ka standard (monic) form kya hai?

y^{''} + p (x) y^{'} + q (x) y = g (x)

Aisi ODE homogeneous kab hoti hai?

Jab forcing term

g (x) \equiv 0

ho.

Superposition principle state karo.

Agar

y_{1}, y_{2}

L [y] = 0

solve karte hain, toh

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

sabhi constants

c_{1}, c_{2}

ke liye

L [y] = 0

solve karta hai.

Non-homogeneous ODEs ke liye superposition kyun fail hota hai?

L [c_{1} y_{1} + c_{2} y_{2}] = (c_{1} + c_{2}) g

, jo

g

ke barabar sirf tab hota hai jab

c_{1} + c_{2} = 1

; free scaling allowed nahi hai.

L [y] = g

ka general solution structure kya hai?

y = y_{h} + y_{p} = c_{1} y_{1} + c_{2} y_{2} + y_{p}

(general homogeneous + ek particular).

Exactly do arbitrary constants kyun?

Second derivative ⇒ "do integrations" ⇒ do constants, jo do initial conditions se fix hote hain.

y_{1}, y_{2}

ka Wronskian define karo.

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

(

[[y_{1}, y_{2}], [y_{1}^{'}, y_{2}^{'}]]

ka determinant).

W (x_{0}) \neq = 0

aapko kya batata hai?

y_{1}, y_{2}

linearly independent hain (aur fundamental set banate hain agar dono ek hi ODE solve karte hain).

Abel's theorem state karo.

W (x) = W (x_{0}) exp (- \int_{x_{0}}^{x} p (t) d t)

; toh

W

ya toh identically zero hai ya kabhi zero nahi.

Kya ek point par

W = 0

dependence ka proof hai?

Generally nahi; sirf ek hi linear ODE ke solutions ke liye (aur tab bhi poore interval par check karna hoga).

Existence–uniqueness condition state karo.

Agar

p, q, g

interval

I ∋ x_{0}

par continuous hain, toh IVP ka

I

par ek unique solution hai.

Operator

L [y] = y^{''} + p y^{'} + q y

ki key property kya hai?

Yeh linear hai:

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

Recall Feynman: 12-saal ke bachche ko explain karo

Ek jhule ki imagine karo. Jhulne ke sirf do "basic" tarike hain (yeh $y_{1}$ aur $y_{2}$ hain). Aur koi bhi jhulna jo aap chahte ho woh in do basics ko mix karke mil sakta hai — ek par do baar push karo, doosre par aadha. Yeh mixing superposition hai. Ab agar koi jhule ko lagaataar push karta rahe (yeh forcing $g$ hai), toh pehle ek tarika dhundo jisme woh settle hota hai ( $y_{p}$ ), phir uske upar free basic swings add karo. Wronskian sirf ek quick check hai ki aapke do basic swings sach mein alag hain aur secretly ek hi swing ke disguise nahi hain.

Equation	Linear?	Homogeneous?
$y^{''} + 3 y^{'} + 2 y = 0$	✅	✅
$y^{''} + x^{2} y = cos x$	✅	❌ (forcing $cos x$ )
$y^{''} + y y^{'} = 0$	❌ ( $y \cdot y^{'}$ product)	—
$y^{''} + sin (y) = 0$	❌ ( $sin y$ nonlinear)	—