Thursday, 14 April 2016

calculus - How can I deduce the value of frac1sqrt4pitintinftyinftysin(y)efrac(xy)24tdy without actually evaluating it?




How can I deduce that
14πtsin(y)e(xy)24tdy=etsin(x)
without actually evaluating the definite integral?


Answer



I will assume the following result:




The solution to the Heat Equation




dfdt=2f



with initial condition f(x,0)=g(x) can be written



f(x,t)=14πte(xy)24tg(y)dy




Now by inserting




f(x,t)=etsin(x)



into the Heat Equation we find that it does satisfy it with the initial condition f(x,0)=sin(x). From the result above it therefore follows that



etsin(x)=14πte(xy)24tsin(y)dy


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