Homogeneous Equation for Mixed Cauchy Problem
We try to solve homogeneous equation for mixed Cauchy problem with null boundary condition of a special form for the parabolic heat equation
with initial condition
and boundary condition
Look for a nonzero solution of the form u(x,t)=X(x)T(t).
We get T'(t)X(x)-T(t)X"(x)=0 or
Second equation has a solution of the form
Equation (*) has solution
Look now for a solution of our equation
From we have
In a particular case all you have to do is to evaluate for the given function.