Solve the following Heat equation:
$u_t=u_{xx}, \quad 0<x<1, t>0$
$u(t,0)=1=u(t,1), \quad t\geq 0$
$u(0,x)=1-\sin(\pi x) \quad 0\leq x\leq 1$
What I have tried:
$u(t,x)=T(t)X(x)$
$\Rightarrow T'X=X''T \quad \Rightarrow \space \frac{T'}{T}=\frac{X''}{X}=-\lambda$
$T'+\lambda T=0$
$X''+\lambda X=0$
solving 1: $T(t)=c_1 e^{-\lambda t}$
solving 2: $X(x)=c_2 \cos(\sqrt{\lambda}x)+c_3\sin(\sqrt{\lambda}x)$
with $u(t,0)=1 \quad \Rightarrow \space c_2=1$
Now I don't know what to do with $u(t,1)=1=\cos(\sqrt{\lambda})+c_3\sin(\sqrt{\lambda})$