Help with Differential equation solution requried Hi guys I am doing some differential equations and I got this one:
I have no idea how he went from (38) to (39). When I solve it by integrating factor, the equation I have is:
$$c_{A}{(t)}=\frac{{c_{af}}}{k\tau+1}+c_{A0}e^{-(\frac{1}{\tau}+k)t}$$
however they have additional $e^{blabla}$ 
It is this website:http://jbrwww.che.wisc.edu/home/jbraw/chemreacfun/ch4/slides-matbal.pdf Page 57 as you can see. Am I missing something?
 A: We have
$$c_A'(t)=\frac1\tau (c_{Af}-c_A(t))-kc_A(t)=\frac1\tau c_{Af}-\left(k+\frac1\tau\right)c_A(t)$$
We can therefore write
$$t=\int_0^t\frac{1}{\frac1\tau c_{Af}-\left(k+\frac1\tau\right)c_A(t')}\,dc_A(t') \tag 1$$
Evaluating the integral in $(1)$ reveals
$$t=\frac{\log\left(\frac1\tau c_{Af}-\left(k+\frac1\tau\right)c_A(0)\right)-\log\left(\frac1\tau c_{Af}-\left(k+\frac1\tau\right)c_A(t)\right)}{k+\frac1\tau} \tag 2$$
Solving $(2)$ for $c_A(t)$ yields
$$\begin{align}
c_A(t)&=\frac{\frac1\tau c_{Af}-\left(\frac1\tau c_{Af}-\left(k+\frac1\tau\right)c_A(0)\right)e^{-\left(k+\frac1\tau\right)t}}{k+\frac1\tau}\\\\
&=c_{A}(0)e^{-\left(k+\frac1\tau\right)t}+\frac{c_{Af}}{k\tau +1}\left(1-e^{-\left(k+\frac1\tau\right)t}\right)\\\\
&=c_{A0}e^{-\left(k+\frac1\tau\right)t}+\frac{c_{Af}}{k\tau +1}\left(1-e^{-\left(k+\frac1\tau\right)t}\right)
\end{align}$$
as was to be shown!
A: Well, your simple integrating factor method actually works. You just have to determine the constant correctly with the initial condition:
$c_A(0)=c_{A0}=\frac{c_{af}}{k\tau+1}+Ke^0$, now solve for $K$
