Differensial Tenglamalar

Differensial tenglamalar — noma'lum funksiya va uning hosilalarini o'z ichiga olgan tenglamalar. Bu robotika, dronlar va raketalar uchun eng muhim matematik vosita, chunki barcha harakat qonunlari differensial tenglamalar orqali ifodalanadi.

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1. Asosiy Tushunchalar

Differensial tenglama nima?

Oddiy differensial tenglama (ODE) — bir o'zgaruvchili funksiya va uning hosilalarini bog'laydigan tenglama.

$$F(t, y, y', y'', \ldots, y^{(n)}) = 0$$

Misol: Erkin tushish tenglamasi

$$\frac{d^2 x}{dt^2} = -g$$

Tartib (Order)

Tenglamadagi eng yuqori hosila tartibini tenglama tartibi deyiladi.

Tenglama	Tartib
$\frac{dy}{dt} = ky$	1-tartib
$\frac{d^2y}{dt^2} + \omega^2 y = 0$	2-tartib
$\frac{d^3y}{dt^3} = f(t)$	3-tartib

Chiziqli va Nochiziqli

Chiziqli ODE — noma'lum funksiya va hosilalar faqat 1-darajada:

$$a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \ldots + a_1(t)y' + a_0(t)y = f(t)$$

Nochiziqli ODE — yuqori darajalar yoki murakkab ifodalar mavjud:

$$\frac{dy}{dt} = y^2, \quad \frac{d^2\theta}{dt^2} = -\sin\theta$$

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2. Birinchi Tartibli ODE

2.1 Ajraladigan O'zgaruvchilar

Agar tenglamani quyidagi ko'rinishga keltirish mumkin bo'lsa:

$$g(y)\,dy = f(t)\,dt$$

Yechish: Har ikki tomonni integrallaymiz.

Misol: $\frac{dy}{dt} = ky$ (eksponensial o'sish/so'nish)

$$\frac{dy}{y} = k\,dt$$ $$\ln|y| = kt + C$$ $$y(t) = Ae^{kt}$$

Qo'llanilishi: Radioaktiv parchalanish, populyatsiya o'sishi, kondensator zaryadsizlanishi.

2.2 Chiziqli Birinchi Tartibli ODE

Standart ko'rinish:

$$\frac{dy}{dt} + P(t)y = Q(t)$$

Yechish usuli: Integrallashtiruvchi ko'paytiruvchi

$$\mu(t) = e^{\int P(t)\,dt}$$

Tenglamani $\mu(t)$ ga ko'paytirib:

$$\frac{d}{dt}[\mu(t)y] = \mu(t)Q(t)$$

Misol: RC zanjir

$$\frac{dV_C}{dt} + \frac{1}{RC}V_C = \frac{V_{in}}{RC}$$

2.3 Boshlang'ich Shart Masalalari (IVP)

Boshlang'ich shart berilganda yagona yechim topiladi:

$$\frac{dy}{dt} = f(t,y), \quad y(t_0) = y_0$$

Misol: $\frac{dy}{dt} = 2t$, $y(0) = 3$

$$y = t^2 + C$$ $$y(0) = 3 \Rightarrow C = 3$$ $$y(t) = t^2 + 3$$

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3. Ikkinchi Tartibli Chiziqli ODE

3.1 Doimiy Koeffitsientli Gomogen Tenglama

$$ay'' + by' + cy = 0$$

Yechish: Xarakteristik tenglama tuzamiz:

$$ar^2 + br + c = 0$$

Uchta holat:

Diskriminant	Ildizlar	Umumiy yechim
$b^2 - 4ac > 0$	$r_1 \neq r_2$ (real)	$y = C_1e^{r_1 t} + C_2e^{r_2 t}$
$b^2 - 4ac = 0$	$r_1 = r_2 = r$	$y = (C_1 + C_2 t)e^{rt}$
$b^2 - 4ac < 0$	$r = \alpha \pm i\beta$	$y = e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t)$

3.2 Garmonik Ossilyator

$$\frac{d^2 x}{dt^2} + \omega^2 x = 0$$

Xarakteristik tenglama: $r^2 + \omega^2 = 0 \Rightarrow r = \pm i\omega$

Yechim:

$$x(t) = A\cos(\omega t) + B\sin(\omega t) = C\cos(\omega t + \phi)$$

Fizik ma'no: Ideal prujinali tebranish, mayatnik (kichik burchaklar).

3.3 So'nuvchi Tebranishlar

$$m\frac{d^2 x}{dt^2} + c\frac{dx}{dt} + kx = 0$$

Standart ko'rinish:

$$\frac{d^2 x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0$$

Bu yerda:

• $\omega_n = \sqrt{k/m}$ — tabiiy chastota
• $\zeta = \frac{c}{2\sqrt{km}}$ — so'nish koeffitsienti (damping ratio)

$\zeta$ qiymati	Holat	Xarakteristika
$\zeta < 1$	Kam so'nuvchi (underdamped)	Tebranadi, asta-sekin so'nadi
$\zeta = 1$	Kritik so'nuvchi (critical)	Eng tez muvozanatga qaytadi
$\zeta > 1$	O'ta so'nuvchi (overdamped)	Sekin muvozanatga qaytadi

Kam so'nuvchi yechim ($\zeta < 1$):

$$x(t) = Ae^{-\zeta\omega_n t}\cos(\omega_d t + \phi)$$ $$\omega_d = \omega_n\sqrt{1-\zeta^2}$$

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4. Nogomogen ODE

4.1 Umumiy Yechim Tuzilishi

$$ay'' + by' + cy = f(t)$$

Umumiy yechim:

$$y = y_h + y_p$$

• $y_h$ — gomogen tenglama yechimi (umumiy)
• $y_p$ — xususiy yechim (particular solution)

4.2 Aniqlanmagan Koeffitsientlar Usuli

$f(t)$ ko'rinishiga qarab $y_p$ ni taxmin qilamiz:

$f(t)$	$y_p$ taxmini
$P_n(t)$ (polinom)	$A_nt^n + A_{n-1}t^{n-1} + \ldots + A_0$
$e^{\alpha t}$	$Ae^{\alpha t}$
$\cos(\beta t)$ yoki $\sin(\beta t)$	$A\cos(\beta t) + B\sin(\beta t)$
$e^{\alpha t}\cos(\beta t)$	$e^{\alpha t}(A\cos\beta t + B\sin\beta t)$

Misol: Majburiy tebranish

$$\frac{d^2 x}{dt^2} + \omega_n^2 x = F_0\cos(\omega t)$$

$y_p = A\cos(\omega t)$ deb taxmin qilib, koeffitsientlarni topamiz.

4.3 Rezonans

Agar majburiy chastota $\omega$ tabiiy chastota $\omega_n$ ga teng bo'lsa:

$$x(t) = \frac{F_0}{2\omega_n}t\sin(\omega_n t)$$

Amplituda cheksiz o'sadi — bu rezonans hodisasi.

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5. Laplace Transformatsiyasi

5.1 Ta'rif

$$\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st}f(t)\,dt$$

5.2 Asosiy Xossalar

$f(t)$	$F(s) = \mathcal{L}\{f(t)\}$
$1$	$\frac{1}{s}$
$t$	$\frac{1}{s^2}$
$t^n$	$\frac{n!}{s^{n+1}}$
$e^{at}$	$\frac{1}{s-a}$
$\sin(\omega t)$	$\frac{\omega}{s^2+\omega^2}$
$\cos(\omega t)$	$\frac{s}{s^2+\omega^2}$

5.3 Hosilalar Transformatsiyasi

$$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$$ $$\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$$

5.4 ODE Yechish Algoritmi

1. Tenglamaning har ikki tomoniga Laplace transformatsiyasi qo'llash
2. $Y(s)$ ni topish (algebraik tenglama)
3. Teskari Laplace transformatsiyasi: $y(t) = \mathcal{L}^{-1}\{Y(s)\}$

Misol: $y'' + 4y = 0$, $y(0) = 1$, $y'(0) = 0$

$$s^2Y(s) - s \cdot 1 - 0 + 4Y(s) = 0$$ $$Y(s)(s^2 + 4) = s$$ $$Y(s) = \frac{s}{s^2 + 4}$$ $$y(t) = \cos(2t)$$

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6. State-Space Ko'rinishi

6.1 Yuqori Tartibli ODE ni 1-Tartibli Sistemaga O'tkazish

Har qanday $n$-tartibli ODE ni $n$ ta birinchi tartibli ODE sistemasiga o'zgartirish mumkin.

Misol: $y'' + 3y' + 2y = u(t)$

O'zgaruvchilar kiritamiz:

$$x_1 = y, \quad x_2 = y'$$

Sistema:

$\dot{x}_1 = x_2$

$\dot{x}_2 = -2x_1 - 3x_2 + u$

6.2 Matritsa Ko'rinishi

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u$$ $$y = \mathbf{C}\mathbf{x} + Du$$

Yuqoridagi misol uchun:

$\mathbf{A} = [[0, 1], [-2, -3]]$, $\mathbf{B} = [0, 1]^T$, $\mathbf{C} = [1, 0]$

Ahamiyati: Boshqarish nazariyasida (control theory) state-space ko'rinish asosiy vosita.

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7. Raqamli Yechish Usullari

Analitik yechim topib bo'lmaydigan hollarda raqamli usullar qo'llaniladi.

7.1 Euler Usuli

$$y_{n+1} = y_n + h \cdot f(t_n, y_n)$$

Bu yerda $h$ — qadam (step size).

Xatolik: $O(h)$ — birinchi tartib aniqlik.

7.2 Runge-Kutta 4-Tartib (RK4)

$$y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)$$

Bu yerda:

$$\begin{aligned} k_1 &= f(t_n, y_n) \\ k_2 &= f(t_n + h/2, y_n + hk_1/2) \\ k_3 &= f(t_n + h/2, y_n + hk_2/2) \\ k_4 &= f(t_n + h, y_n + hk_3) \end{aligned}$$

Xatolik: $O(h^4)$ — ancha aniqroq.

7.3 Python Kodi

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

def damped_oscillator(y, t, zeta, omega_n):
    x, v = y
    dxdt = v
    dvdt = -2*zeta*omega_n*v - omega_n**2*x
    return [dxdt, dvdt]

# Parametrlar
zeta = 0.3
omega_n = 2.0
y0 = [1.0, 0.0]  # x(0)=1, v(0)=0
t = np.linspace(0, 10, 500)

# Yechish
sol = odeint(damped_oscillator, y0, t, args=(zeta, omega_n))

plt.plot(t, sol[:, 0])
plt.xlabel('Vaqt (s)')
plt.ylabel('x(t)')
plt.title("So'nuvchi tebranish")
plt.grid(True)
plt.show()

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8. Barqarorlik (Stability)

8.1 Muvozanat Nuqtasi

$\dot{y} = f(y)$ uchun muvozanat nuqtasi: $f(y^*) = 0$

8.2 Chiziqli Sistemalar Barqarorligi

$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}$ sistemasi barqaror agar $\mathbf{A}$ ning barcha xos qiymatlari manfiy real qismga ega bo'lsa.

Xos qiymatlar	Barqarorlik
Barchasi $\text{Re}(\lambda) < 0$	Asimptotik barqaror
Birortasi $\text{Re}(\lambda) > 0$	Nobarqaror
$\text{Re}(\lambda) = 0$	Marjinal barqaror

8.3 Fizik Talqin

• Barqaror: Tizim muvozanatga qaytadi (masalan, so'nuvchi mayatnik)
• Nobarqaror: Tizim muvozanatdan uzoqlashadi (teskari mayatnik)

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9. Qo'llanilishi

Robotika

• Robot qo'li dinamikasi
• PID kontrollerlar
• Traektoriya rejalashtirish

Dronlar

• Parvoz dinamikasi (6DOF model)
• Barqarorlik tahlili
• Kalman filtri

Raketalar

• Traektoriya tenglamalari
• Yonilg'i sarfi
• Orbital mexanika

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Xulosa

Mavzu	Asosiy g'oya
1-tartib ODE	Ajratish, integrallashtiruvchi ko'paytiruvchi
2-tartib ODE	Xarakteristik tenglama
Laplace	Algebraik usulga o'tkazish
State-space	Matritsa ko'rinishi
Raqamli usullar	Euler, RK4
Barqarorlik	Xos qiymatlar tahlili

Keyingi qadam: Masalalar orqali mustahkamlash va Python/MATLAB da amaliyot.