# Homogeneous Equation for Mixed Cauchy Problem

Submitted by Structure on Sun, 11/18/2007 - 19:13.

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.