Suppose $v$ solves
PDE: $\quad v_{t} = v_{xx} + t^2 x(x-1) , \quad 0 < x < 1$ and $t > 0$.
BC: $\quad v(0,t) = 0, \quad v(1,t) = 0$
IC: $\quad v(x,0) = 0$

Suppose $w$ solves
PDE: $\quad w_{t} = w_{xx}, \quad 0 < x < 1$ and $t > 0$.
BC: $\quad w(0,t) = \sin(t), \quad w(1,t) = te^t$
IC: $\quad w(x,0) = 0$

Suppose $z$ solves
PDE: $\quad z_{t} = z_{xx} , \quad 0 < x < 1$ and $t > 0$.
BC: $\quad z(0,t) = 0, \quad z(1,t) = 0$
IC: $\quad z(x,0) = x(x-1)$

a) We want to solve:
PDE: $\quad u_{t} = u_{xx} + 3 t^2 x(x-1) , \quad 0 < x < 1$ and $t > 0$.
BC: $\quad u(0,t) = 6\sin(t), \quad u(1,t) = 6 te^t$
IC: $\quad u(x,0) = 8 x(x-1)$

Write $u$ in terms of $v$, $w$, and $z$
$u(x,t) = {}$

b) How about:
(notice changes in both BC and IC, you might have to solve two more problems)
PDE: $\quad u_{t} = u_{xx} + 3 t^2 x(x-1) , \quad 0 < x < 1$ and $t > 0$.
BC: $\quad u(0,t) = 1+6\sin(t), \quad u(1,t) = 6 te^t$
IC: $\quad u(x,0) = (1-x)+8 x(x-1) + 9 \sin(\pi x)$

$u(x,t) = {}$

help (formulas)