Consider the following PDE. Where $u(x,t) = X(x)T(t)$
$u_{tt}+u_t = u_{xx}$
$u(0,t)=u(\pi,t)=0$
$u(x,0)=0$
$u_t(x,0)=10$
I am having trouble solving for my $T(t)$, it comes down to an ODE
So just to save some time and work, for my eigenfunctions, I got $X_n(x) = \sin(nx)$ with $\lambda = n$.
$T''+T'+n^2T = 0$
$T(0)=0$
$T'(0)=10$
Now I am having trouble writing my solution as hyperbolic functions. My roots from auxilarily equation is $r=\dfrac{-1 \pm \sqrt{1-4n^2}}{2}$. Any idea on how to get rid of that $\frac{1}{2}$ in front?