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Unformatted text preview: Math 415 Quiz 8A Given the heat conduction equation in two space dimensions, 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 2 ( X 00 Y T + XY 00 T ) = XY T , 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 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,- , and we immediately have t s differential equation: T + 2 T = 0. There are a few equivalent ways to continue with the equation X 00 Y + XY 00 =- XY , one of which is to write this as X 00 X =- Y 00- Y Y = , 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 equationson different variables which are always equal to each other....
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