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Q8A_130

# Q8A_130 - Math 415 Quiz 8A Given the heat conduction...

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Math 415 Quiz 8A Given the heat conduction equation in two space dimensions, α 2 ( u xx + u yy ) = u t , and assuming that u ( x, y, t ) = X ( x ) Y ( y ) T ( t ), find ordinary differential equations satisfied by X ( x ), Y ( y ), and T ( t ). Plugging u = XY T into the differential equation gives us α 2 ( X 00 Y T + XY 00 T ) = XY T 0 , and noting that the left hand side has T as a factor, we can separate the equation as X 00 Y + XY 00 XY = T 0 α 2 T . The left side of this equation does not depend on t at all, and the right side depends only on t , so we can let both sides equal a constant, - λ , and we immediately have t ’s differential equation: T 0 + α 2 λT = 0. There are a few equivalent ways to continue with the equation X 00 Y + XY 00 = - λXY , one of which is to write this as X 00 X = - Y 00 - λY Y = μ, where, we have set both sides equal to a constant for the usual reason that we have two functions dependent on different variables which are always equal to each other. This gives us the differential equations T 0 + α 2 λT = 0 , X 00 - μX = 0 , Y 00 + ( λ + μ ) Y = 0 .

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