Problem with riccati equation

Hi every body, I have a big problem with an riccati equation, and I want to solve it in MATLAB, can anyone help me? Thank you;

dq(t)/dt= A*q^2(t)+ B*q(t)+C;

Wher A,B and C are known

Answer 1

[Walter Roberson]

Do you have the symbolic toolbox?

If your q^2(t) is intended to mean q(t)^2 then the solution is

q(t) = 1/2*(-B+tan(1/2*(4*C*A-B^2)^(1/2)*(t+C1))*(4*C*A-B^2)^(1/2))/A

where C1 is the arbitrary constant of integration whose value could be pinned down if you had a boundary condition.

If, however, q^2(t) is intended to indicate d^2q/dt the second order differential of q with respect to t, then the solution would be different,

q(t) = exp(1/2/A*(1+(1-4*B*A)^(1/2))*t)*C2 + exp(-1/2*(-1+(1-4*B*A)^(1/2))/A*t)*C1-C/B

where C1 and C2 are constants of integration.

Answer 2

[Imène ALLAB]

thank you very much Walter!! but how did you do that?? I can't do it with ode45 ? i forget to montion that I have a finale condition q(T) :( , is that the same withe this condition?

(and yes it means q(t)^2))

Thanks again :) have a good day.

[Walter Roberson]

What is the boundary condition?

If your boundary condition is q(T) = Q for some constant T and some constant Q, then the constant of integration C1 that I noted would equal to

-(T*(4*C*A-B^2)^(1/2) - 2*arctan((B+2*Q*A)/(4*C*A-B^2)^(1/2))) / (4*C*A-B^2)^(1/2)

I solved this using Maple's dsolve()

dsolve(diff(q(t),t)= A*q(t)^2 + B*q(t)+C)

If you had the symbolic toolbox, you could do very much the same thing using MuPad's dsolve() command.