Alright — let’s break this down carefully. ## 1. Understanding the problem We have a metal plate in the first quadrant \( x \ge 0, y \ge 0 \). Boundary conditions: - Edge along the y-axis (\(x = 0, y \ge 0\)): \(T(0, y) = 0\) (Dirichlet, fixed at zero). - Edge along the x-axis (\(y = 0, x \ge 0\)): given an initial temperature distribution: \[ T(x, 0) = \begin{cases} 100(4 - x), & 0 < x < 4, \\ 0, & x > 4 \end{cases} \] The other two edges (as \( x \to \infty \) or \( y \to \infty \)) have no explicit boundary conditions but we expect \(T \to 0\) far away in a steady state. The plate’s edges along \(x=0\) and \(y=0\) are given, and we are to find the steady state temperature distribution \(T(x,y)\). --- ## 2. Steady-state heat equation For steady-state with no internal heat sources in 2D: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] (Laplace’s equation in \(x, y\)). Boundary conditions: \[ T(0,y) = 0 \quad \text{for } y > 0, \] \[ T(x,0) = f(x), \] with \(f(x) = 100(4-x)\) for \(0 < x < 4\), 0 otherwise. Also, we expect \(T \to 0\) as \(x \to \infty\) or \(y \to \infty\) (physically reasonable for an infinite quarter plane with zero temp on \(x=0\) and finite temp on \(y=0\)). --- ## 3. Method of solution for quarter plane We can solve Laplace’s equation in the quarter-plane \(x \ge 0, y \ge 0\) with \(T(0,y)=0\) using an odd extension in \(x\) to the full plane, then using Poisson’s formula for the half-plane \(y \ge 0\). Let’s first extend \(f(x)\) to \(x<0\) as an odd function about \(x=0\), so that \(T(0,y)=0\) is automatically satisfied for the half-plane problem in \(y>0\) (whole \(x\) line) with boundary \(T(x,0) = f_{\text{odd}}(x)\). --- The odd extension of \(f(x)\): Original \(f(x)\) for \(x \ge 0\): \(f(x) = 100(4-x)\) for \(0 \le x \le 4\), \(0\) for \(x>4\). Define \(f_{\text{odd}}(x)\) for all \(x\): \[ f_{\text{odd}}(x) = \begin{cases} 100(4-x), & 0 \le x \le 4, \\ 0, & x > 4, \\ -100(4 + x), & -4 \le x < 0, \\ 0, & x < -4. \end{cases} \] Check: At \(x=0\), \(f_{\text{odd}}(0) = 400\), but wait — the physical boundary \(x=0\) requires \(T=0\), so