## 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
and

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

(*)We get
(**)
Second equation has a solution of the form

So
Equation (*) has solution

or
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.