📘 partial differential equations

1. **Problem Statement:** We want to solve the heat conduction equation $$4 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}$$ for $$0 < x < \frac{\pi}{2}$$ and $$

1. **Problem Statement:** Solve the wave equation $$\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2}$$ with $$c^2 = 1$$ and initial displacement

1. **Problem statement:** Solve the first order partial differential equation (PDE) given by: $$3U_x - 2U_y + U = 2x + 1$$

1. **Énoncé du problème :** Montrer que l'équation aux dérivées partielles (E) : $$x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \fra

1. **State the problem:** We need to find the complete integral of the first-order PDE given by:

1. **Problem Statement:** Solve the PDEs using the method of separation of variables. (a) $$\frac{\partial u}{\partial x} + u = \frac{\partial u}{\partial t}$$ with initial conditi

1. **Problem Statement:** Solve the PDEs given in 3a and 3b. 2. **Recall the PDEs:**

1. **Problem Statement:** Solve the partial differential equation (PDE) $$\frac{\partial^2 z}{\partial y \partial x} = x^4 y$$ with initial conditions $$z(x,0) = x^4$$ and $$z(1,y)

1. **Problem Statement:** We want to solve the heat equation for a rod of length $L$ with zero temperature at both ends and an initial temperature distribution $f(x)$. The problem

1. **State the problem:** We need to determine the X-trics (characteristics) of the PDE given by $$z - p^2 = z$$ and find the particular integral surface passing through the parabo

1. **Problem statement:** Solve the 2D heat equation $$\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)$$

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 $$u(x,0) = x$$.