I must solve this initial boundary value problem by transforming it into homogeneous BCs.

$\displaystyle \[\\(PDE)\; u_{t}=\alpha ^{2}u_{xx} \: \: \: \: \: \: \; \; \;\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 0<x<1\\

\\

$

$\displaystyle (BCs)\; \left\{\begin{matrix}

u(0,t)=1) & 0<t< \infty \\

u_{x}(1,t)+hu(1,t)=1 & 0<t< \infty

\end{matrix}\right.\\$

$\displaystyle (IC)\;\: \: u(x,0)=sin(\pi x)+x\; \: \: \: \: \: \: \: \: \: \: 0\leq x\leq 1 $

---> My work

I set $\displaystyle \[s(x,t)=a(t)x+b(t)(1-x)+U(x,t)\]

$ where S(x,t) is the steady state and U(x,t) is the transient

So i substitute s(x,t) into the BCs

$\displaystyle \[s(0,t)=1

$

$\displaystyle s_{x}(1,t)+hs(1,t)=1$

These two equations can give usb(t)=1 and a(t)=2/1-h?

and therefore u(x,t)=2x/(1-h) + (1-x) + U(x,t) =1 - (1+h/1-h)x + U(x,t)

MY PROBLEM is, how did they get b(t)=1 and a(t) = 2/1-h.

I know i have have to make s(0,t) = to something likea(t) + b(t) = 1and s_x(1,t) =?something?so that i can plug it into the $\displaystyle \[

s_{x}(1,t)+hs(1,t)=1\]

$ but I am not sure what s and s_x should be.

Can someone please tell me what they are so I can finally get a(t) and b(t).

Then I can finally finish solving this problem.

Thanks in advance!