Solve $\lambda u_t+u_x=-u$ with $x,t > 0$ and $u(x,0)=0$ and $u(0,t)=\sinh{t}$ using method of characteristics I am asked to solve the following PDE using the method of characteristics:
$\lambda u_t+u_x=-u$ with $x,t > 0$ and $u(x,0)=0$ and $u(0,t)=\sinh{t}$

So I get (I use q instead of t given that t is already used): 
$t_q=\lambda$ with $t(0,s)=s$ so $t(q)=\lambda q + s$

$x_q=1$ with $x(0,s)=0$ so $x(q)=q+0$
and 
$u_q=-u$ with $u(0,s)=0$ so $u(q)=e^{-q}-1$ (?) 
Using $u(0,t)=\sinh{t}$, we get $u(q,t)=e^{-q}-1+\sinh{t}$ (?)
We also know that $q=x$ thus $s=t-\lambda x$, using the first and second ODEs. So our final result is $u(x,t)=e^{-x}-1+\sinh{t}$.
Now, I am used to have only one initial condition, not two of them. (Also, I don't see why they use $t$ while we generally use $y$ instead.) So I am not sure if this is really correct, and would like your review.
Thanks for your help !
 A: $$\lambda u_t+u_x=-u \quad\text{with}\quad \begin{cases} x,t > 0 \\ u(x,0)=0 \\ u(0,t)=\sinh{t}\end{cases}$$
Charpit-Lagrange system of characteristic equations :
$$\frac{dt}{\lambda}=\frac{dx}{1}=\frac{du}{-u}$$
First characteristic equation from $\frac{dt}{\lambda}=\frac{dx}{1}$ :
$$t-\lambda x=c_1$$
Second characteristic equation from $\frac{dx}{1}=\frac{du}{-u}$ :
$$ue^x=c_2$$
General solution on the form of implicit equation $c_2=F(c_1)=ue^x=F(t-\lambda x)$ :
$$u=e^{-x}F(t-\lambda x)$$
The function $F(X)$ is arbitrary, to be determined according to the specified conditions.
Boundary condition : $u(0,t>0)=\sinh(t)=e^0F(t-0)$ determines the function $$F(X)=\sinh(X) \qquad X>0$$
Initial condition : $u(x>0,0)=0=e^{-x}F(-\lambda x)$ determines the function $$F(X)=0\quad\begin{cases} X>0 \text{ if } \lambda <0 \\ X<0 \text{ if } \lambda >0 \end{cases}$$
All together $F(X)$ is a piecewise function , excluding the case $\lambda<0$.
$$F(X)=\begin{cases} \sinh(X) && X>0 \\0 && X<0\end{cases}$$
In the general solution $X=t-\lambda x$ thus :
$$u(x,t)=\begin{cases} e^{-x}\sinh(t-\lambda x) && t>\lambda x \\0 && t<\lambda x\end{cases}\qquad x,t,\lambda>0$$

ADDITION, in order to answer to a comment.
Copy from https://en.wikipedia.org/wiki/Method_of_characteristics

