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Solution: Heat equation The flow of heat along a thin
Chapter 12, Problem 69E(choose chapter or problem)
Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)
\(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\)
where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1.
\(u(x, t)=4 e^{-4 t} \cos 2 x\)
Questions & Answers
QUESTION:
Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)
\(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\)
where u is a measure of the temperature at a location x on the bar at time! and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1.
\(u(x, t)=4 e^{-4 t} \cos 2 x\)
ANSWER:Solution 69EStep1:In the Given problem Heat equation The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) where u is a measure of the temperature at a location x on the bar at time t and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heat equation with k = 1.u(x, t) = 4e-4t cos 2x