Problem with riccati equation
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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
回答（2 个）
Walter Roberson
2011-8-30
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.
Imène ALLAB
2011-8-30
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Walter Roberson
2011-8-30
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.
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2011-8-30
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