4.6.13 · D1Case 3 - complex conjugate roots — Euler's formula connecti
Jab koi bouncing, wiggling system (ek spring, ek swinging door, ek electric circuit) ko ek equation se describe kiya jaata hai, to uski natural motion hamesha ek shrinking-ya-growing wave hoti hai: ek spinning arrow (woh hai i e α x α β i
Yeh page kuch bhi assume nahi karta. Har woh squiggle jo parent note use karta hai, neeche khol ke rebuild kiya gaya hai, ek aisi order mein jahan har piece usse pehle wale piece par khada ho.
Pehle kuch bhi samjhne se pehle, symbol y ′
y y ′ y ′′
y function hai: ek machine jo input x y ( x ) x y y ′ derivative hai: har point par us curve ki steepness (slope). Socho woh direction jisme ek marble roll karta.y ′′ curviness , ya physics mein acceleration.
Intuition Topic ko derivatives ki zaroorat kyun haiSpring ka law kehta hai "acceleration depend karta hai position par". Acceleration hai y ′′ y hi hai equation a y ′′ + b y ′ + cy = 0 y ′′ y ′
Dekho prerequisite Characteristic Equation of Linear ODEs jisme bataya gaya hai ki ye primes m
Definition Second-order linear homogeneous ODE with constant coefficientsNaam ko tod ke dekho:
ODE = ordinary differential equation: ek equation jo ek function ko uske apne derivatives ke saath mix karti hai.Second-order : sabse bada prime jo present hai woh y ′′ Linear : y , y ′ , y ′′ y 2 sin y y ⋅ y ′ Homogeneous : right-hand side 0 Constant coefficients : a , b , c x
Letters:
a , b , c ∈ R a , b , c R a = 0 y ′′
Definition Characteristic rootm guess karte hain ki solution y = e m x e m x m y ′ = m e m x y ′′ = m 2 e m x e m x a m 2 + bm + c = 0. m characteristic root hai. Socho: har valid m
± ek saath do answers ka shorthand hai — ek + −
Δ = b 2 − 4 a c Δ discriminant hai: quadratic formula mein square root ke neeche wali quantity. Yeh decide karta hai ki kitne, aur kis tarah ke roots milenge.
Δ positive negative bilkul bhi nahi hai. Woh single sign flip hi humein majboor karta hai ek naye tarah ka number invent karne ko — agla section.
Jab Δ < 0
Definition Imaginary uniti i define kiya jaata hai ==i 2 = − 1 − 9 = 9 ⋅ − 1 = 3 i complex number ki shape hoti hai α + i β α i β Socho: ise number line par mat rakho — ise ek plane par rakho. α β 2-D plane mein ek point hai (origin se ek arrow).
i i 0 = 1 , i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 , … Socho: i 9 0 ∘
α − i β α + i β conjugate imaginary part ka sign flip karta hai: α − i β Socho: horizontal (real) axis ke across mirror-reflection. Ek real equation ke complex roots hamesha ek mirror pair ke roop mein aate hain — yehi "α ± i β
α β m = α ± i β
α = − 2 a b real part hai — envelope number. Socho: wave kitni tezi se shrink ya grow karti hai.β = 2 a 4 a c − b 2 > 0 imaginary part hai (sirf size) — frequency number. Socho: wave kitni tezi se wiggle karti hai.
Yeh bas roots se padhe gaye do real numbers ke naam hain.
e imaginary power tak uthana (e i θ Maclaurin Series of exp, sin, cos se build hua aur Euler's Formula and the Unit Circle mein khola gaya.
cos θ sin θ 1 θ
cos θ horizontal (x) coordinate.sin θ vertical (y) coordinate.
Socho: jaise jaise θ 1 cos 2 θ + sin 2 θ = 1
Intuition Sine aur cosine kyun, koi aur wiggle kyun nahii ek hi rhythm hai. Woh coincidence hai Euler's formula e i θ = cos θ + i sin θ 1 i
C 1 , C 2 solutions ka koi bhi weighted sum phir bhi ek solution hoga — yeh hai Superposition Principle for Linear ODEs . Unknown weights C 1 , C 2 arbitrary constants hain jo baad mein starting conditions (initial values) se fix hote hain.
W W u , v W = u u ′ v v ′ = u v ′ − v u ′ . W = 0 linearly independent hain — genuinely alag, ek doosre ki copy nahi — to yeh ek complete basis form karte hain. Poori detail Wronskian and Linear Independence mein. Socho: W = 0
Derivatives y prime y double prime
The ODE a y'' + b y' + c y = 0
Guess e to the m x gives quadratic in m
Discriminant Delta = b squared - 4 a c
Roots alpha plus or minus i beta
Euler formula e to i theta
Sine cosine on unit circle
Real solution e to alpha x times cos and sin
Khud test karo — daayein side cover karo.
y ′′ derivative ka derivative — y
Hamare ODE ke liye "homogeneous" ka kya matlab hai? right-hand side 0
y e m x kyunki e m x a m 2 + bm + c = 0
Δ = b 2 − 4 a c Δ < 0
i imaginary unit jisme i 2 = − 1
i complex plane mein 9 0 ∘
α + i β α − i β
m = α ± i β α growth/decay envelope e α x
β oscillation frequency (period 2 π / β
cos θ sin θ horizontal (x) aur vertical (y) respectively.
∣ e i θ ∣ = 1 kyunki cos 2 θ + sin 2 θ = 1 1
W = 0 do solutions linearly independent hain aur ek valid basis form karte hain.
