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?
Note that $n \geq 1$. Hence, $1-4n^2 < 0$. Let $1-4n^2 = -k_n^2$. This gives us that $r = -\dfrac12 \pm i\dfrac{k_n}2$. Then $T_n(t)$ is of the form $$T_n(t) = \exp(-t/2) \left(A_n \cos(k_nt/2) + B_n \sin(k_n t/2) \right)$$ $T_n(0) = 0 \implies A_n = 0$. Hence, $$T_n(t) = B_n \exp(-t/2) \sin(k_n t/2)$$ $$T_n'(t) = -1/2 B_n \exp(-t/2) \sin(k_n t/2) + B_n k_n/2 \exp(-t/2) \cos(k_n t/2)$$ $$T_n'(0) = B_n k_n/2 = 10 \implies B_n = \dfrac{20}{k_n}$$ Hence, $$T_n(t) = \exp(-t/2) \dfrac{20 \sin(k_n t/2)}{k_n}$$
EDIT
Your final solution is now of the form $$u(x,t) = 20 \exp(-t/2) \left(\sum_{n=1}^{\infty} \dfrac{\sin(k_n t/2) \sin(nx)}{k_n} \right)$$
EDIT
If you get the roots as $r=a,b \in \mathbb{R}$, then the solution is of the form $$c_1 \exp(at) + c_2 \exp(bt)$$ which can be rewritten as $$\exp \left(\dfrac{a+b}2t \right) \left(k_1 \cosh \left(\dfrac{(a-b)t}2 \right) + k_2 \sinh \left(\dfrac{(a-b)t}2 \right) \right)$$