How to solve this differential equation $m\ddot z + b\dot z + mg =0 $? I was trying to do my physic homework and I got stuck here:
$$m\ddot z + b\dot z + mg =0 $$
We know that $z(0) =z_0$ and $\dot z(0)=0$.
It is also suggested to use $\tau \equiv \frac m b$ 
I've added the teacher's answer below. 

 A: Feel free to point out any errors as the morning coffee hasn't kicked in.
Divide through by $m$ and use the substitution $\tau = \frac{b}{m}$:
$\ddot{z}+\tau \dot{z}+g=0$
Then use the substitution $v=\dot{z}$.  Then $\dot{v}=\ddot{z}$
$\dot{v}+\tau v+g=0$
$\dot{v}+\tau v=-g$
Now that you have a first order ODE, just use an integrating factor: $\mu(t)=e^{\int \tau dt}=e^{\tau t}.$
$e^{\tau t}\dot{v}+e^{\tau t}\tau v=-g$  
$(e^{\tau t}v)'=-e^{\tau t}g$
Integrate both sides w.r.t. $t$
$v=\frac{-e^{\tau t}g}{\tau}+c$
$\dot{z}=\frac{-e^{\tau t}g}{\tau}+c$
Integrating once more w.r.t $t$:
$z=\frac{-e^{\tau t}g}{\tau^2}+ct+d$
To find $c$ and $d$, plug in the conditions given.
A: Let's solve the ODE,
$\displaystyle \ddot x + a\dot x+b=0$ where $a,b$ are arbitrary real constants. It's a homogeneous $2^{\text{nd}}$ order differential equation which has a corresponding A.E(Auxiliary Equation) given by,
$\displaystyle x^2+ax+b=0$ , Let it's roots be real and distinct (which is evident here as $b^2\ne 4m^2 g$) and denoted by $\alpha,\beta$.
So the solution of the ODE is given by $\displaystyle x(t)=c_1e^{\alpha t}+c_2e^{\beta t}$
In this case $\displaystyle a=\frac{b}{m},b=g$ and boundary conditions are $z(0)=z_0,\dot z(0)=0$ plugging in which we find ,
$\displaystyle z(t)=\frac{z_0}{\beta-\alpha}\left(\beta e^{\alpha t}-\alpha e^{\beta t}\right)$ 
where $\displaystyle \alpha,\beta = \frac{-b\pm\sqrt{b^2-4gm}}{2m}$
A: We found the answer. 
You just have to do as @emka said, then once you got:
$$ \dot u +\frac {1}{\tau} u=0$$ 
you know that the general answer will be :
$$ u(t) = C e^{(-\frac {t}{\tau} )}$$
You find $C = -\tau $ then you'll have:
$$ \dot z(t) = \tau g (e^{(-\frac {t}{\tau})}-1)$$
Then all you have to do is integrate from 0 to t, and use you conditions to get the final equation.
