How to Solve a Ricatti Equation?

[CIMP]

Dear all,

Someone could help me to understand how I can resolve the following equation :

dV/dt= A V^2 + B V + C

Where V :V(t), A(t), B(t), C(t)

Is there any method or indications about this ?

Thanks in advance,

Indira

 

[Mark44]

Dear all,

Someone could help me to understand how I can resolve the following equation :

dV/dt= A V^2 + B V + C

Where V :V(t), A(t), B(t), C(t)

Is there any method or indications about this ?

Thanks in advance,

Indira

Separate the variables.

Rewrite the equation as
dV/(A V^2 + B V + C) = dt, and then integrate.

Is this a homework problem?

 

[LCKurtz]

Dear all,

Someone could help me to understand how I can resolve the following equation :

dV/dt= A V^2 + B V + C

Where V :V(t), A(t), B(t), C(t)

Is there any method or indications about this ?

Thanks in advance,

Indira

Separate the variables.

Rewrite the equation as
dV/(A V^2 + B V + C) = dt, and then integrate.

I don't think that will work if the A, B, and C are functions of t as the OP indicates. If C(t) is identically 0 it might be solvable as a Bernoulli equation. In general there's no hope for an explicit solution.

 

[Mark44]

Sorry, I totally missed it that A, B, and C were functions of t.

 

[CIMP]

Thanks a lot Mark44 and LCKurtz.

C is also function of time. In that case, if there is not hope for explicit solution, should I try to solve it numerically? and if yes? which method could I use?

I know how varies A, B and C as a function of another variable "Z". In other words:

dV/dt= A(Z) V^2 + B(Z) V + C(Z)


Thanks,

I:)

 

[Talonz]

You are basically solving it for V as a differential equation, is that correct? Wolfram Alpha calls it a Riccati equation. The answer looks eerily similar to the quadratic formula, with an arctan involved, so that might be a hint as to how to go about solving a portion of it.

 

[D H]

You are basically solving it for V as a differential equation, is that correct? Wolfram Alpha calls it a Riccati equation. The answer looks eerily similar to the quadratic formula, with an arctan involved, so that might be a hint as to how to go about solving a portion of it.

Sometimes you have to be very explicit with Mathematica. This is one of those times. You should have told Mathematica that z is a function of t: http://www.wolframalpha.com/input/?...V/dt=+a(z(t))*V^2+++b(z(t))*V+++c(z(t))+for+V

Yes, this is a Riccati equation, and no, there isn't a general closed-form solution. There are special cases for which a closed-form solution is known.

 

[Talonz]

I know how varies A, B and C as a function of another variable "Z". In other words:)

I thought he had said z here was a independent variable, and not a function of t? (edit) Nevermind... Different versions of the same formula floating around, I looked at the last one.

 

[JJacquelin]

Thanks a lot Mark44 and LCKurtz.

C is also function of time. In that case, if there is not hope for explicit solution, should I try to solve it numerically? and if yes? which method could I use?

I know how varies A, B and C as a function of another variable "Z". In other words:

dV/dt= A(Z) V^2 + B(Z) V + C(Z)


Thanks,

I:)

dV/dt= A(Z) V^2 + B(Z) V + C(Z)

if Z is a function of t there is no known analytical method to solve it (except in some particular forms of functions A(Z(t)), B(Z(t)), C(Z(t)). You have to solve it numerically.

if Z is not function of t, then it can be analyticaly solved :
t = integral of dV/(A V² + B V + C)
For a given value of Z which dosn't depends on t, A(Z), B(Z), C(Z) are constants.
So you obtain t as a function of V and Z
Inverting this fonction would lead to V as a function of t and Z
But if, afterwards, you make Z varying with t, the result would be false.

 

[CIMP]

Dear All,

Thanks a lot to everybody for the advices. I learn a lot in this forum always...

Yes! I see that this is a Ricatti equation and I think the best is to solve it with numerical methods... asssuming the initial conditions for each one the variables depending on "t".

Thanks again,

I:)