📘 differential equations

1. **Problem Statement:** We want to express the velocity vector $v$ of a swimmer in a canal flowing upward with speed $k$, where the swimmer swims at speed $c$ in still water, alw

1. The problem gives the differential equation $$x(y^2 + 13) \, dx + y(x^2 + 6) \, dy = 0$$ and asks to analyze it. 2. We are given a potential function candidate: $$\frac{x^2 y^2}

1. **Problem Statement:** Solve the differential equation $$y' - 2y = \sin(3t)$$ using Laplace transforms.

1. مسئله: حل معادله دیفرانسیل $$x^2 y'' + 2xy' + xy = 0$$ به روش سری حول نقطه $$x=0$$. 2. روش سری: فرض می‌کنیم جواب به صورت سری توانی باشد:

1. **Problem Statement:** The population of mosquitoes in a certain environment follows the differential equation:

1. **State the problem:** Solve the differential equation $\left(x^2D^2 + xD\right)y = 0$, where $D = \frac{d}{dx}$. 2. **Rewrite the equation:** The operator $D$ represents differ

1. **Problem:** Solve the differential equation $$\sec x \frac{dy}{dx} = y^2 - y$$ with initial condition $$y(0) = 4$$. 2. **Rewrite the equation:** Multiply both sides by $$\cos x

1. **Stating the problem:** We are given a differential equation $$\frac{dy}{dx} = -5x^3 - 23x^2 + 10x - 20$$ with initial condition $$y(0) = 2$$. We need to compute $$y_1, y_2, y_

1. **Problem statement:** Solve the differential equation $$y'' - y' - 6y = 0$$ with initial conditions $$y(0) = 1$$ and $$y'(0) = -1$$ using Laplace transform. 2. **Formula and ru

1. Мәселені айқындау: Берілген дифференциалдық теңдеу $$y'(x) = 4 \int_0^x (t - x) y(t) \, dt + 3 \cos x$$

1. **Problem 1: Find the general solution of the differential equation** $$tds = (3t + 1)sdt + t^3 e^{3t} dt$$

1. The problem is to solve the first differential equation given: $$\frac{dy}{dx} + x^2 y^2 = x$$. 2. This is a first-order nonlinear ordinary differential equation because of the

1. Muammo: Berilgan tizimning yechimini topish kerak: $$9y'' + 4y''' + 2y' + 8y = 2u'' + 8u'' + 4u' + 2u$$, bu yerda $y$ chiqish, $u$ kirish. 2. Avvalo, tenglamani soddalashtiramiz

1. **Problem:** Find the Laplace Transform of the function \( f(t) = e^{-4t} \cos 3t + e^{-3t} t^3 \). 2. **Formula and rules:**

1. **State the problem:** Solve the differential equation $$3y'' + y' - 4y = x \sin x$$. 2. **Identify the type of equation:** This is a nonhomogeneous linear second-order differen

1. مسئله: حل معادله دیفرانسیل برنولی $y' + \frac{y}{x} = x^4 y^3$. 2. معادله برنولی به شکل کلی است: $y' + P(x)y = Q(x)y^n$ که در اینجا $P(x) = \frac{1}{x}$، $Q(x) = x^4$ و $n=3$.

1. مسئله: حل معادله دیفرانسیل برنولی به شکل $$y' + \frac{y}{x} = x^r y^r$$ است. 2. فرمول و روش حل: معادله برنولی به صورت کلی $$y' + P(x)y = Q(x)y^n$$ است. برای حل، ابتدا متغیر جدید

1. **State the problem:** Solve the differential equation $$y'' - 2y' + y = (x+5)^2$$. 2. **Identify the type of equation:** This is a non-homogeneous linear second-order different

1. **Problem Statement:** Solve the differential equation $$\frac{dy}{dt} = -\frac{y}{t} + 2$$ for the general solution. 2. **Identify the type of equation:** This is a first-order

1. State the problem. We are asked to find the Laplace transform of $3e^{-2t}$.

1. **Problem Statement:** Solve the differential equation $ (2x - y) dx - (x + 4y) dy = 0 $ and find the implicit solution. 2. **Identify the type:** This is a first order differen