📘 partial differential equations

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:

1. **Problem 1: Solve for A1, A2, A3, A4 in the Laplace equation with given boundary values and uniform mesh spacing.** Given the Laplace equation $$\frac{\partial^2 U}{\partial x^

1. Problem i) Solve $$\frac{\partial^2 z}{\partial x^2} = xy$$ by direct integration. Step 1: Integrate once with respect to $x$:

1. **Problem (a):** Find the temperature $u(x,t)$ in an 80 cm insulated bar with initial temperature $100\sin\left(\frac{\pi x}{80}\right)$ and ends at 0°C. - The heat equation is

1. The problem is to analyze and transform the partial differential equation (PDE) $$\frac{\partial^2 u}{\partial x^2} + e^{2x} \frac{\partial^2 u}{\partial y^2} + y \frac{\partial

1. Problem: Solve the PDE $$z = px + qy + \sqrt{1 + p^2 + q^2}$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. Step 1: Recognize this is a firs

1. We are asked to solve the partial differential equation (PDE): $$(z^2 - 2yz - y^2)p + (xy + zx)q = xy - zx.$$ 2. Here, $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partia

1. **Classify the PDEs and find their types** - For equation $U_{xx} - U_{yy} = 0$