Masalalar — Differensial Tenglamalar

35 ta masala: osondan qiyinga tartiblangan.

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Boshlang'ich (1-10)

Masala 1: Ajraladigan o'zgaruvchilar

Tenglamani yeching:

$$\frac{dy}{dx} = 3x^2$$

Boshlang'ich shart: $y(0) = 2$

Yechim

$dy = 3x^2\,dx$ $y = x^3 + C$ $y(0) = 2 \Rightarrow C = 2$ $\boxed{y = x^3 + 2}$

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Masala 2: Eksponensial o'sish

Bakteriya populyatsiyasi $\frac{dP}{dt} = 0.5P$ qonuniga bo'ysunadi. Agar $P(0) = 100$ bo'lsa, $P(4)$ ni toping.

Yechim

$P(t) = P_0 e^{kt} = 100e^{0.5t}$ $P(4) = 100e^{2} = 100 \cdot 7.389 \approx \boxed{739}$

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Masala 3: Radioaktiv parchalanish

Radioaktiv modda yarim parchalanish davri $T_{1/2} = 5$ yil. Parchalanish konstantasini toping.

Yechim

$N(t) = N_0 e^{-\lambda t}$ $N(T_{1/2}) = \frac{N_0}{2} = N_0 e^{-\lambda \cdot 5}$ $\frac{1}{2} = e^{-5\lambda}$ $\ln(0.5) = -5\lambda$ $\boxed{\lambda = \frac{\ln 2}{5} \approx 0.1386 \text{ yil}^{-1}}$

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Masala 4: Birinchi tartib chiziqli

Tenglamani yeching:

$$\frac{dy}{dx} + 2y = 4$$

Yechim

Integrallashtiruvchi ko'paytiruvchi: $\mu = e^{\int 2\,dx} = e^{2x}$

$e^{2x}\frac{dy}{dx} + 2e^{2x}y = 4e^{2x}$ $\frac{d}{dx}[e^{2x}y] = 4e^{2x}$ $e^{2x}y = 2e^{2x} + C$ $\boxed{y = 2 + Ce^{-2x}}$

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Masala 5: Kondensator zaryadsizlanishi

RC zanjirda $\frac{dV}{dt} + \frac{V}{RC} = 0$, $R = 1000\,\Omega$, $C = 10\,\mu F$. Agar $V(0) = 5V$, $V(0.02s)$ ni toping.

Yechim

$\tau = RC = 1000 \cdot 10 \times 10^{-6} = 0.01\,s$

$V(t) = V_0 e^{-t/\tau} = 5e^{-t/0.01}$ $V(0.02) = 5e^{-2} = 5 \cdot 0.135 \approx \boxed{0.68\,V}$

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Masala 6: Xarakteristik tenglama

Tenglamaning umumiy yechimini toping:

$$y'' - 5y' + 6y = 0$$

Yechim

Xarakteristik tenglama: $r^2 - 5r + 6 = 0$ $(r-2)(r-3) = 0 \Rightarrow r_1 = 2, r_2 = 3$

$\boxed{y = C_1 e^{2x} + C_2 e^{3x}}$

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Masala 7: Takrorlanuvchi ildizlar

$y'' - 4y' + 4y = 0$

Yechim

$r^2 - 4r + 4 = 0 \Rightarrow (r-2)^2 = 0 \Rightarrow r = 2$ (ikki marta)

$\boxed{y = (C_1 + C_2 x)e^{2x}}$

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Masala 8: Kompleks ildizlar

$y'' + 4y = 0$

Yechim

$r^2 + 4 = 0 \Rightarrow r = \pm 2i$

$\alpha = 0$, $\beta = 2$

$\boxed{y = C_1\cos(2x) + C_2\sin(2x)}$

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Masala 9: Garmonik ossilyator

Prujinali sistema $m = 2\,kg$, $k = 8\,N/m$. Tabiiy chastotani va davrni toping.

Yechim

$\omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{8}{2}} = 2\,rad/s$ $T = \frac{2\pi}{\omega_n} = \frac{2\pi}{2} = \boxed{\pi \approx 3.14\,s}$

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Masala 10: Boshlang'ich shartli masala

$y'' + 9y = 0, \quad y(0) = 0, \quad y'(0) = 6$

Yechim

Umumiy yechim: $y = C_1\cos(3x) + C_2\sin(3x)$

$y(0) = C_1 = 0$

$y' = -3C_1\sin(3x) + 3C_2\cos(3x)$ $y'(0) = 3C_2 = 6 \Rightarrow C_2 = 2$

$\boxed{y = 2\sin(3x)}$

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O'rta (11-22)

Masala 11: So'nish koeffitsienti

Tizim: $\ddot{x} + 4\dot{x} + 20x = 0$. $\omega_n$, $\zeta$, va $\omega_d$ ni toping.

Yechim

Standart ko'rinish: $\ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 x = 0$

$\omega_n^2 = 20 \Rightarrow \omega_n = \sqrt{20} = 2\sqrt{5} \approx 4.47\,rad/s$

$2\zeta\omega_n = 4 \Rightarrow \zeta = \frac{4}{2 \cdot 2\sqrt{5}} = \frac{1}{\sqrt{5}} \approx 0.447$

$\omega_d = \omega_n\sqrt{1-\zeta^2} = 2\sqrt{5}\sqrt{1-0.2} = 2\sqrt{5}\sqrt{0.8} = 2\sqrt{4} = \boxed{4\,rad/s}$

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Masala 12: Kritik so'nish

$m\ddot{x} + c\dot{x} + kx = 0$ sistemada $m = 1\,kg$, $k = 100\,N/m$. Kritik so'nish uchun $c$ qiymatini toping.

Yechim

Kritik so'nishda $\zeta = 1$: $c_{cr} = 2\sqrt{km} = 2\sqrt{100 \cdot 1} = \boxed{20\,Ns/m}$

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Masala 13: Nogomogen tenglama

$y'' + y = 2\cos(x)$

Yechim

$y_h = C_1\cos x + C_2\sin x$

Chunki $\cos x$ gomogen yechimga kiradi, $y_p = x(A\cos x + B\sin x)$

$y_p' = A\cos x + B\sin x + x(-A\sin x + B\cos x)$ $y_p'' = -2A\sin x + 2B\cos x - x(A\cos x + B\sin x)$

$y_p'' + y_p = -2A\sin x + 2B\cos x = 2\cos x$

$B = 1$, $A = 0$

$\boxed{y = C_1\cos x + C_2\sin x + x\sin x}$

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Masala 14: Laplace transformatsiyasi

$\mathcal{L}\{t \cdot e^{-2t}\}$ ni hisoblang.

Yechim

$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$

$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$

$\mathcal{L}\{t\} = \frac{1}{s^2}$

$\mathcal{L}\{te^{-2t}\} = \frac{1}{(s+2)^2}$

$\boxed{\frac{1}{(s+2)^2}}$

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Masala 15: Laplace bilan yechish

Laplace usuli bilan yeching: $y' + 3y = 6$, $y(0) = 0$

Yechim

$sY(s) - y(0) + 3Y(s) = \frac{6}{s}$ $Y(s)(s+3) = \frac{6}{s}$ $Y(s) = \frac{6}{s(s+3)}$

Parchalab: $\frac{6}{s(s+3)} = \frac{A}{s} + \frac{B}{s+3}$

$6 = A(s+3) + Bs$

$s=0: A = 2$, $s=-3: B = -2$

$Y(s) = \frac{2}{s} - \frac{2}{s+3}$ $\boxed{y(t) = 2 - 2e^{-3t} = 2(1 - e^{-3t})}$

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Masala 16: State-space o'tkazish

$\ddot{x} + 5\dot{x} + 6x = u(t)$ ni state-space ko'rinishga o'tkazing.

Yechim

$x_1 = x$, $x_2 = \dot{x}$

$$\dot{x}_1 = x_2, \quad \dot{x}_2 = -6x_1 - 5x_2 + u$$

$\mathbf{A} = [[0, 1], [-6, -5]]$, $\mathbf{B} = [0, 1]^T$

Javob: $\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u$

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Masala 17: Xos qiymatlar

$\mathbf{A} = [[0, 1], [-2, -3]]$ matritsasining xos qiymatlarini va barqarorligini aniqlang.

Yechim

$\det(\mathbf{A} - \lambda\mathbf{I}) = 0$

$\lambda^2 + 3\lambda + 2 = 0$

$(\lambda + 1)(\lambda + 2) = 0$

$\lambda_1 = -1$, $\lambda_2 = -2$

Ikkala xos qiymat manfiy, shuning uchun asimptotik barqaror.

$\boxed{\lambda = -1, -2 \text{ (barqaror)}}$

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Masala 18: Euler usuli

$y' = y$, $y(0) = 1$ ni Euler usuli bilan $h = 0.5$ qadam bilan $t = 1$ gacha yeching.

Yechim

$y_{n+1} = y_n + h \cdot f(t_n, y_n) = y_n + 0.5 \cdot y_n = 1.5y_n$

$y_0 = 1$ $y_1 = 1.5 \cdot 1 = 1.5$ ($t = 0.5$) $y_2 = 1.5 \cdot 1.5 = 2.25$ ($t = 1$)

Aniq yechim: $y(1) = e^1 \approx 2.718$

$\boxed{y(1)_{Euler} = 2.25 \text{ (xatolik } \approx 17\%\text{)}}$

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Masala 19: RK4 bir qadam

$y' = -2y$, $y(0) = 1$, $h = 0.1$ uchun RK4 ning birinchi qadamini hisoblang.

Yechim

$f(t, y) = -2y$

$k_1 = f(0, 1) = -2$ $k_2 = f(0.05, 1 + 0.05 \cdot (-2)) = f(0.05, 0.9) = -1.8$ $k_3 = f(0.05, 1 + 0.05 \cdot (-1.8)) = f(0.05, 0.91) = -1.82$ $k_4 = f(0.1, 1 + 0.1 \cdot (-1.82)) = f(0.1, 0.818) = -1.636$

$y_1 = 1 + \frac{0.1}{6}(-2 + 2(-1.8) + 2(-1.82) + (-1.636))$ $y_1 = 1 + \frac{0.1}{6}(-10.876) = 1 - 0.1813 = 0.8187$

Aniq: $y(0.1) = e^{-0.2} = 0.8187$

$\boxed{y(0.1) \approx 0.8187 \text{ (juda aniq!)}}$

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Masala 20: Uzatish funksiyasi

$2\ddot{y} + 6\dot{y} + 4y = u(t)$, nol boshlang'ich shartlar bilan. Uzatish funksiyasi $H(s) = \frac{Y(s)}{U(s)}$ ni toping.

Yechim

Laplace transformatsiyasi: $2s^2Y(s) + 6sY(s) + 4Y(s) = U(s)$ $Y(s)(2s^2 + 6s + 4) = U(s)$

$H(s) = \frac{Y(s)}{U(s)} = \frac{1}{2s^2 + 6s + 4} = \boxed{\frac{1}{2(s+1)(s+2)}}$

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Masala 21: Teskari mayatnik

Teskari mayatnik uchun chiziqlashtrilgan tenglama: $\ddot{\theta} - \frac{g}{L}\theta = 0$. Nima uchun nobarqaror?

Yechim

Xarakteristik tenglama: $r^2 - \frac{g}{L} = 0$

$r = \pm\sqrt{\frac{g}{L}}$

Bitta ildiz musbat ($+\sqrt{g/L}$), shuning uchun yechim $e^{+\sqrt{g/L} \cdot t}$ ni o'z ichiga oladi va cheksizga intiladi.

$\boxed{\text{Musbat xos qiymat mavjud} \Rightarrow \text{nobarqaror}}$

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Masala 22: Rezonans chastotasi

$\ddot{x} + 0.4\dot{x} + 4x = F_0\cos(\omega t)$. Rezonans yaqinida maksimal amplituda qaysi chastotada?

Yechim

$\omega_n = 2$, $\zeta = \frac{0.4}{2 \cdot 2} = 0.1$

Rezonans chastotasi (so'nish bilan): $\omega_r = \omega_n\sqrt{1 - 2\zeta^2} = 2\sqrt{1 - 0.02} = 2\sqrt{0.98} \approx \boxed{1.98\,rad/s}$

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Qiyin (23-35)

Masala 23: Tizim barqarorligi

Boshqarish tizimi $\mathbf{A}$ matritsasi bilan berilgan (3x3): qatorlar $[0,1,0]$, $[0,0,1]$, $[-6,-11,-6]$.

Xos qiymatlarni toping va barqarorlikni aniqlang.

Yechim

Xarakteristik polinom: $\det(\mathbf{A} - \lambda\mathbf{I}) = -\lambda^3 - 6\lambda^2 - 11\lambda - 6 = 0$ $\lambda^3 + 6\lambda^2 + 11\lambda + 6 = 0$

Faktorlash: $(\lambda + 1)(\lambda + 2)(\lambda + 3) = 0$

$\lambda_1 = -1$, $\lambda_2 = -2$, $\lambda_3 = -3$

$\boxed{\text{Barcha } \lambda < 0 \Rightarrow \text{asimptotik barqaror}}$

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Masala 24: Dron parvoz dinamikasi

Quadcopter balandlik dinamikasi (soddashtirilgan): $m\ddot{z} = T - mg - k_d\dot{z}$

$m = 1\,kg$, $g = 10$, $k_d = 0.5$. Barqaror balandlik uchun talab qilinadigan itarish kuchini toping.

Yechim

Barqaror holatda $\ddot{z} = 0$, $\dot{z} = 0$: $0 = T - mg - 0$ $T = mg = 1 \cdot 10 = \boxed{10\,N}$

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Masala 25: PID boshqaruv tenglamasi

PID kontrollerli sistema: $G(s) = \frac{1}{s^2 + 2s + 1}$, $K_p = 10$.

Yopiq sikl uzatish funksiyasini toping (faqat P boshqaruv).

Yechim

$T(s) = \frac{K_p G(s)}{1 + K_p G(s)} = \frac{\frac{10}{s^2+2s+1}}{1 + \frac{10}{s^2+2s+1}}$

$T(s) = \frac{10}{s^2 + 2s + 1 + 10} = \boxed{\frac{10}{s^2 + 2s + 11}}$

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Masala 26: Sensor fusion model

IMU sensorida akselerometr va giroskopdan keladigan ma'lumotlar: $\dot{\theta}_{gyro} = \omega + n_g$ $\theta_{accel} = \theta + n_a$

Bu yerda $n_g$, $n_a$ shovqinlar. Kalman filtri uchun state va o'lchov tenglamalarini yozing.

Yechim

State tenglamasi: $\dot{\theta} = \omega$ Diskret: $\theta_{k+1} = \theta_k + \omega_k \cdot \Delta t + w_k$

O'lchov tenglamasi: $z_k = \theta_k + v_k$

$\boxed{\mathbf{F} = 1, \quad \mathbf{H} = 1, \quad \mathbf{B} = \Delta t}$

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Masala 27: Raketa uchish tenglamasi

Raketa (vertikal uchish, havo qarshiligi bilan): $m\frac{dv}{dt} = T - mg - \frac{1}{2}\rho v^2 C_D A$

Bu nochiziqli tenglamani $v$ kichik bo'lganda chiziqlashtiring.

Yechim

$v$ kichik bo'lganda $v^2 \approx 0$:

$m\frac{dv}{dt} \approx T - mg$ $\frac{dv}{dt} = \frac{T - mg}{m} = \frac{T}{m} - g$

$\boxed{v(t) = \left(\frac{T}{m} - g\right)t + v_0}$

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Masala 28: Orbital tezlik

Orbital mexanikadan: $\ddot{r} - r\dot{\theta}^2 = -\frac{GM}{r^2}$

Doiraviy orbitada ($r = const$) orbital tezlikni toping.

Yechim

$r = const \Rightarrow \ddot{r} = 0$

$-r\dot{\theta}^2 = -\frac{GM}{r^2}$ $\dot{\theta}^2 = \frac{GM}{r^3}$

Orbital tezlik $v = r\dot{\theta}$: $v^2 = r^2\dot{\theta}^2 = r^2 \cdot \frac{GM}{r^3} = \frac{GM}{r}$

$\boxed{v = \sqrt{\frac{GM}{r}}}$

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Masala 29: Robot qo'li dinamikasi

Oddiy robot qo'li (bir bo'g'in): $I\ddot{\theta} + b\dot{\theta} + mgl\sin\theta = \tau$

Kichik burchaklar uchun chiziqlashtiring ($\sin\theta \approx \theta$) va uzatish funksiyasini toping.

Yechim

$I\ddot{\theta} + b\dot{\theta} + mgl\theta = \tau$

Laplace: $Is^2\Theta(s) + bs\Theta(s) + mgl\Theta(s) = T(s)$

$\frac{\Theta(s)}{T(s)} = \boxed{\frac{1}{Is^2 + bs + mgl}}$

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Masala 30: Traektoriya tenglamasi

Gorizontal otilgan jism: $\ddot{x} = 0$, $\ddot{y} = -g$, $x(0) = 0$, $y(0) = h$, $\dot{x}(0) = v_0$, $\dot{y}(0) = 0$.

$y(x)$ traektoriyasini toping.

Yechim

$x(t) = v_0 t \Rightarrow t = \frac{x}{v_0}$

$y(t) = h - \frac{1}{2}gt^2$

$y = h - \frac{1}{2}g\left(\frac{x}{v_0}\right)^2 = \boxed{h - \frac{g}{2v_0^2}x^2}$

Parabola!

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Masala 31: Van der Pol ossilyatori

Nochiziqli tenglama: $\ddot{x} - \mu(1 - x^2)\dot{x} + x = 0$

$x = 0$ atrofida chiziqlashtiring ($x$ kichik).

Yechim

$x$ kichik bo'lganda $x^2 \approx 0$:

$\ddot{x} - \mu\dot{x} + x = 0$

Xarakteristik: $r^2 - \mu r + 1 = 0$

$r = \frac{\mu \pm \sqrt{\mu^2 - 4}}{2}$

$\mu > 0$ uchun musbat real qism mavjud $\Rightarrow$ nobarqaror.

$\boxed{\text{Muvozanat nobarqaror, limit cycle hosil bo'ladi}}$

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Masala 32: Ko'p darajali tizim

Sistema: $\dot{x}_1 = x_2$, $\dot{x}_2 = -x_1 - 2x_2 + u$

Uzatish funksiyasini toping ($y = x_1$).

Yechim

Laplace: $sX_1 = X_2$ $sX_2 = -X_1 - 2X_2 + U$

$X_2 = sX_1$ ni ikkinchisiga qo'yib: $s(sX_1) = -X_1 - 2sX_1 + U$ $s^2X_1 + 2sX_1 + X_1 = U$ $X_1(s^2 + 2s + 1) = U$

$\frac{Y}{U} = \frac{X_1}{U} = \boxed{\frac{1}{(s+1)^2}}$

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Masala 33: Tebranish amplitudasi

So'nuvchi tebranish: $x(t) = 5e^{-0.2t}\cos(4t)$

5 ta to'liq tebranishdan keyin amplituda qanchaga tushadi?

Yechim

Davr: $T = \frac{2\pi}{\omega_d} = \frac{2\pi}{4} = \frac{\pi}{2}$

5 davr vaqti: $5T = \frac{5\pi}{2}$

Amplituda: $A(t) = 5e^{-0.2t}$

$A(5T) = 5e^{-0.2 \cdot 5\pi/2} = 5e^{-0.5\pi} = 5e^{-1.571} = 5 \cdot 0.208 \approx \boxed{1.04}$

Amplituda $\approx 79\%$ kamaydi.

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Masala 34: Routh-Hurwitz

$s^3 + 4s^2 + 5s + 2 = 0$ tenglamasining barcha ildizlari chap yarim tekislikda ekanligini Routh kriteri bilan tekshiring.

Yechim

Routh jadvali:

$s^3$	1	5
$s^2$	4	2
$s^1$	$\frac{4 \cdot 5 - 1 \cdot 2}{4} = \frac{18}{4} = 4.5$	0
$s^0$	2

Birinchi ustun: 1, 4, 4.5, 2 — barchasi musbat.

$\boxed{\text{Ishora o'zgarishi yo'q} \Rightarrow \text{barqaror}}$

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Masala 35: Tsiolkovsky tenglamasi

Raketa: $m\frac{dv}{dt} = -v_e\frac{dm}{dt}$, bu yerda $v_e$ — gazlar chiqish tezligi.

Agar $m_0 = 1000\,kg$, $m_f = 100\,kg$, $v_e = 3000\,m/s$ bo'lsa, $\Delta v$ ni toping.

Yechim

$m\,dv = -v_e\,dm$ $dv = -v_e\frac{dm}{m}$

Integrallaymiz: $\int_0^{\Delta v} dv = -v_e \int_{m_0}^{m_f} \frac{dm}{m}$ $\Delta v = -v_e [\ln m_f - \ln m_0] = v_e \ln\frac{m_0}{m_f}$

$\Delta v = 3000 \cdot \ln\frac{1000}{100} = 3000 \cdot \ln 10 = 3000 \cdot 2.303$

$\boxed{\Delta v \approx 6908\,m/s \approx 6.9\,km/s}$

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Qo'shimcha Masalalar

Amaliy: Python simulyatsiya

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

def rocket(y, t, ve, mdot, g):
    """Raketa harakati (vertikal)"""
    h, v, m = y
    if m <= 100:  # yonilg'i tugadi
        mdot = 0
    thrust = ve * mdot
    dhdt = v
    dvdt = thrust/m - g
    dmdt = -mdot
    return [dhdt, dvdt, dmdt]

# Boshlang'ich shartlar
y0 = [0, 0, 1000]  # h=0, v=0, m=1000 kg
t = np.linspace(0, 100, 1000)

# Parametrlar
ve = 3000  # m/s
mdot = 10  # kg/s
g = 9.81

sol = odeint(rocket, y0, t, args=(ve, mdot, g))

fig, axs = plt.subplots(3, 1, figsize=(10, 8))
axs[0].plot(t, sol[:, 0]/1000)
axs[0].set_ylabel('Balandlik (km)')
axs[1].plot(t, sol[:, 1])
axs[1].set_ylabel('Tezlik (m/s)')
axs[2].plot(t, sol[:, 2])
axs[2].set_ylabel('Massa (kg)')
axs[2].set_xlabel('Vaqt (s)')
plt.tight_layout()
plt.show()

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Natijalar Jadvali

Daraja	Masalalar	Mavzular
Boshlang'ich	1-10	Ajratish, xarakteristik, IVP
O'rta	11-22	So'nish, Laplace, state-space, raqamli
Qiyin	23-35	Tizimlar, dron, raketa, robot

Jami: 35 masala