Solving a PDE: Can Someone Help?

[kaniello]

Hallo,
I ended up with this PDE:

du/dt + A*u² =C

A and C are constants and U is only function of time.
It looks simple but...
Can someone help me?
Thanks a lot in advance

 

[CompuChip]

It is a Riccati equation, and indeed it is slightly more tricky than you may initially think.

 

[kaniello]

Thank you so much! That was a quick answer!

 

[dextercioby]

Can't you just separate variables ?

 

[X89codered89X]

@dextercioby. That would work to solve for a function of u (e.g. f(u) = ...) , but kaniello here is solving for the function u(t) that satisfies the ODE... a much tricker problem that satisfies the riccati equation.

 

[dextercioby]

Hmmm

$$\frac{du}{dt} = C-Au^2 \Rightarrow \int dt = \int\frac{du}{C-Au^2} \Rightarrow...$$

??

 

[kaniello]

..which would be
tanh⁻1($\sqrt{a/c}$ / (a/c).
I am currently studying the article from CompuChip and reference given there. Probably this is the solution that comes out from the Riccati equation setting q1=0, q0 and q2= const.

 

[AlephZero]

@dextercioby. That would work to solve for a function of u (e.g. f(u) = ...) , but kaniello here is solving for the function u(t) that satisfies the ODE... a much tricker problem that satisfies the riccati equation.

It is a Riccati equation, and indeed it is slightly more tricky than you may initially think.

The Riccati equation is only "tricky" in the general case - for example when C in the OP's equation is replaced by something like $Ct^n$

As dextercioby said, this special case is straighforward to integrate.