Help solving a first order non-linear differential equation derived from the navier-stokes equation I am an engineer studying an unsteady-state flow through a pipe.  The transient Bernoulli equation of this system, which I picked up from here (http://higheredbcs.wiley.com/legacy/college/fox/0471742996/webpdf/ch06.pdf) 
yields this differential equation:
$$\frac{dv(t)}{dt}+av^2(t)+bP(t)=c$$
This is a first order non-linear differential equation.  Unfortunately, I have no experience solving non-linear differential equations.  From the research I have done, this type of equation looks similar to the Ricatti equation.  Is there a closed form solution to the above equation?  How can I solve this equation?  I'm interested in getting a function that shows how the pressure or velocity of the system decays with time.
FYI:  v(t) is velocity,  P(t) is pressure,  (a,b,c) are constants.  Thank You all.
 A: $$\frac{dv(t)}{dt}+av^2(t)+bP(t)=c$$
Let $v(t)=\frac{y'(t)}{ay(t)}$
$v'=\frac{y''}{ay}-\frac{y'^2}{ay^2}=-a\left(\frac{y'}{ay} \right)^2-bP(t)+c$
$\frac{y''}{ay}=-bP(t)+c$
$$y''+a(bP(t)-c)y=0$$
Since the form of the function $P(t)$ is not defined, the second order ODE is on a general form. No general solution can be provided. 
So, one can expect to solve it only if the fonction $P(t)$ is known. What is more, expecting doesn't mean to be sure to solve it. Solutions are known only in some particular cases of functions $P(t)$.
As a consequence, in the general case, numerical methods are recommended.
Note :
if you intend to try to solve some equations of the kind :
$$y''+F(t)y=0$$
there are a few guidelines :
$F(t)=c \: \to \: y(t)$ involves sinusoïdal or hyperbolic functions.
$F(t)=a+bt \: \to \: y(t)$ involves Airy functions.
$F(t)=a+bt+ct^2 \: \to \: y(t)$ involves parabolic cylinder functions.
$F(t)=a+\frac{b}{t} \: \to \: y(t)$ involves Bessel functions.
$F(t)=a+\frac{b}{t^2} \: \to \: y(t)$ involves Bessel functions.
$F(t)=at^p \: \to \: y(t)$ involves Bessel functions.
$F(t)=ae^{bt}+c \: \to \: y(t)$ involves Bessel functions.
$F(t)=a+\frac{b}{t}+ \frac{c}{t^2} \: \to \: y(t)$ involves Kummer or Whittaker or Coulomb wave functions.
Etc. These are only rough indications about the kind of functions. Depending on particular values of the parameters, a function might become a function of lower level. 
A: It is in fact a Ricatti equation. To find an explicit solution you need first a particular solution. This can be difficult to find even when an explicit form for $P$ is known.
