Simple ODE Analytical Solution Question: $\frac{dx}{dt} = ax + bu$ my question is in regards to the following formula: 
\begin{align}
\frac{dx}{dt}=ax+bu
\end{align}
where x(0)=0 & u=constant and a & b are scalar
The answer to this or at least what I was given as the answer is the following: 
\begin{align}
x=-\frac{bu}{a}(1-e^{at)}
\end{align}
Now I understand that I need to manipulate the formula and integrate both sides. Utilizing the integration table I perform the following: 
Since u=constant we can then say bu=b therefore:
\begin{align}
\frac {dx}{ax+b}= {dt}
\end{align}
\begin{align}
\int \frac {dx}{ax+b}= \int{dt}
\end{align}
\begin{align}
\frac{1}{a}ln|ax+b| = t + C
\end{align}
\begin{align}
e^{ln|ax+b|} = e^{(t + C)a}
\end{align}
\begin{align}
ax+b = e^{at+Ct}
\end{align}
\begin{align}
x = -\frac{b}{a} \frac{e^{at}e^{Ct}}{a}
\end{align}
I do not see where I am going wrong in the calcs above. I would appreciate any help in identifying where I went wrong. 
Thank you

$$\begin{align}
x = -\frac{b}{a} \frac{e^{at}e^{Ct}}{a}
\end{align}$$
This line isn't correct, it should be: 
$$\begin{align}
x = -\frac{b}{a} +\frac{e^{(t+C)a}}{a}
\end{align}$$
