📘 partial differential equations

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 $U_{xx} - U_{yy} = 0$