📘 partial differential equations

1. **Problem Statement:** We need to solve the homogeneous spherical partial differential equation (PDE):

1. **Problem Statement:** We are given the partial differential equation (PDE) in spherical coordinates:

1. **State the problem:** Solve the partial differential equation (PDE) given by $$y^2 (x - y) p + x^2 (y - x) q = z (x^2 + y^2)$$

1. مسئله: حل معادله دیفرانسیل جزئی $$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} + x^2 + 5x$$ با روش جداسازی متغیرها. 2. روش جداسازی متغیرها: فرض می‌کن

1. **Problem Statement:** Solve the PDE $$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \sin t \cos x \cos y$$

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

1. **State the problem:** We are given the partial differential equation (PDE) $$\frac{\partial u}{\partial x} - 2 \frac{\partial^2 u}{\partial y^2} = 0$$ where $u = u(x,y)$ is an

1. The problem is to write down the general linear partial differential equation (PDE) of second order. 2. A second-order linear PDE involves partial derivatives of a function $u(x

1. The problem is to write down the auxiliary equations used in Charpit's method for solving first-order partial differential equations. 2. Charpit's method transforms a first-orde

1. **Problem:** Show that the family of spheres $$x^2 + y^2 + (z - c)^2 = r^2$$ satisfies the PDE $$y p - x q = 0$$ where $$p = \frac{\partial z}{\partial x}$$ and $$q = \frac{\par

1. **Stating the problem:** We are given the partial differential equation (PDE) $$(y-z)p + (x+y)q = z - x,$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\

1. **Stating the problem:** We are given the partial differential equation (PDE) $$(v - z) p + (x + y) q = z - x$$

1. **Problem statement:** Determine the solution of the initial-value problem for the wave equation on a semi-infinite string with given initial and boundary conditions.

1. **State the problem:** Solve the wave equation for the function $u(x,t) = \sin(x - at)$ where $a$ is a constant. 2. **Recall the wave equation:** The standard one-dimensional wa

1. **Problem Statement:** We have three tasks involving partial differential equations (PDEs):

1. **Problem 1: Eliminate arbitrary constants a and b** (a) Given $$z = ax + by + a^2 b^2$$

1. **Eliminate arbitrary constants a and b:** (a) Given $z = ax + by + a^2 b^2$.

1. **Stating the problem:** We want to solve the partial differential equation (PDE) given by $$z = px + qy + \sqrt{1 + p^2 + q^2}$$ where $p = \frac{\partial z}{\partial x}$ and $

1. **Problem Statement:** We want to solve the partial differential equation (PDE)

1. **Problem:** Verify if $u = e^x \sin 2y$ satisfies the partial differential equation $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 11.$$\n\n2. **Step

1. The problem asks whether the function $f(x,y) = \cos\left(x - \frac{y}{3}\right)$ satisfies Laplace's equation. 2. Laplace's equation in two variables is given by: