1. Find the solution by separating variables method. (20%) (∂ u)/(∂ x) = (∂ u)/(∂ y) B.C.: u(0, y) = 2e^-4y

(a) Please draw the heuristic search tree for solving the famous 8 puzzle problem. The puzzle is contained in a tray that can be thought of as a grid of nine squares. The tray contains only eight tiles, which means that one of the grids is always empty. The tiles are numbered from 1 to 8. The rule of the game is that a tile can be slid into an empty slot. The initial and goal states of an 8-puzzle are shown as the follows. Please draw the heuristic search tree for solving the puzzle.Initial stateGoal state

1. Find the solution by separating variables method. (20%) \[ \frac{\partial u}{\partial x} = \frac{\partial u}{\partial y} \] B.C.: \( u(0, y) = 2e^{-4y} \)

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