📘 partial differential equations

1. **State the problem:** We want to solve the heat equation $$\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \

1. The first question asks to define the complete solution of a partial differential equation (PDE). 2. A complete solution of a PDE is a solution that contains as many arbitrary c

1. The problem is to derive the one-dimensional wave equation, which describes how waves propagate along a string or in a medium. 2. Start with the physical setup: consider a strin

1. **Problem statement:** Derive the one-dimensional heat equation, which models how heat diffuses through a rod over time. 2. **Physical setup:** Consider a thin rod along the $x$

1. The problem is to solve the partial differential equation using Lagrange's method: $$yzp - xzq = xy$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\parti

1. **Постановка задачи:** Дана задача Коши для уравнения с частными производными:

1. **Problem statement:** Solve the heat equation $$\frac{\partial u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2}$$ for an infinite insulated rod with initial condition $$u

1. مسئله: حل معادله گرما $$\frac{\partial u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2}$$ با شرایط اولیه $$u(x,0) = \begin{cases} T_0, & x>0 \\ -T_0, & x<0 \end{cases}$$ د

1. مسئله: حل معادله گرما $$\frac{\partial u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2}$$ با شرایط اولیه $$u(x,0) = \begin{cases} T_0 & x>0 \\ -T_0 & x<0 \end{cases}$$ در

1. مسئله: حل معادله دیفرانسیل جزئی $$\frac{\partial u}{\partial t} = 3 \frac{\partial^2 u}{\partial x^2}$$ با شرایط مرزی $$u(0,t) = 0$$، $$u(2,t) = 0$$ و شرط اولیه $$u(x,0) = x$$.

1. **Problem:** Form a differential equation of all planes which are at a constant distance $a$ from the origin. 2. **Step 1:** Recall the equation of a plane: $$Ax + By + Cz + D =

1. **State the problem:** Solve the partial differential equation (PDE) $$x^2 \frac{\partial^2 z}{\partial x^2} - y^2 \frac{\partial^2 z}{\partial y^2} = xy.$$\n\n2. **Identify the

1. The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. 2. The standard form of the heat equation in one dimensio

1. **State the problem:** Simplify or analyze the expression $uxx + uxxyy + uyy$ where subscripts denote partial derivatives. 2. **Identify notation:** Here, $uxx$ means $\frac{\pa

1. **State the problem:** Solve the partial differential equation (PDE) given by $$x \frac{\partial u}{\partial x} = y \frac{\partial u}{\partial y}$$ where $u = u(x,y)$. 2. **Assu

1. **Problem Statement:** Solve the partial differential equation (PDE) given by $$n \frac{\partial u}{\partial x} = y \frac{\partial u}{\partial y}$$

1. **State the problem:** Solve the partial differential equation (PDE) given by $$n \frac{\partial u}{\partial m} = y \frac{\partial u}{\partial y}$$

1. **Énoncé du problème** : Nous avons l'équation aux dérivées partielles (EDP) suivante :

1. **Problem statement:** Solve the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(x,0) = 3 \sin n\pi x,$$ $$u(0,t

1. **Stating the problem:** Solve the partial differential equation $$\frac{\partial^2 z}{\partial x^2} + z = 0$$ with initial conditions at $$x=0$$: $$z = e^y$$ and $$\frac{\parti

1. **بیان مسئله:** حل معادله دیفرانسیل با مشتقات جزئی لاپلاس در مختصات استوانه‌ای برای تابع دما $T(r,\theta,z)$ که معادله آن به صورت زیر است: