1. Consider the partial differential equation: $\displaystyle q(x) { \partial u\over \partial t}={ \partial \over \partial x} \left( p(x) { \partial u\over \partial x} \right)$ (a) Confirm that $$U(x) = A +(B-A){\int_0^x p(s)^{-1} ds\over \int_0^L p(s)^{-1} ds }$$ Is a solution of the partial differential equation (b) Confirm that if $u=U+v$ then $v$ indeed satisfies the partial differential equation if $u$ does (c) Confirm that $X(x)T(t)$ is indeed a solution of the partial diff eq if $X$ and $T$ are solutions of the following respectively: $${d\over dx} \left(p(x) {dX\over dx}\right) – kq(x)X = 0$$ $${dT\over dt} = kT$$ where $k$ is a constant

1.  Consider the partial differential equation:

$\displaystyle q(x) { \partial u\over \partial t}={ \partial \over \partial x} \left( p(x) { \partial u\over \partial x} \right)$

(a)   Confirm that

$$U(x) = A +(B-A){\int_0^x p(s)^{-1} ds\over \int_0^L p(s)^{-1} ds }$$

Is a solution of the partial differential equation

(b)   Confirm that if $u=U+v$ then $v$ indeed satisfies the partial differential equation if $u$ does

(c)   Confirm that $X(x)T(t)$ is indeed a solution of the partial diff eq if $X$ and $T$ are solutions of the following respectively:

$${d\over dx} \left(p(x) {dX\over dx}\right) – kq(x)X = 0$$

$${dT\over dt} = kT$$

where $k$ is a constant

SOLUTION ===============