How can I integrate symbolically a coupled vector function?

Hello,

I am trying to get a symbolic expression for the integration of a vector function arrising from a coupled differential equation, given that I want a symbolic result, I can't use ode45 and the int()m function is not working for me. My code is the following, if anyone could give me a hint about how to integrate it, it would be really appreciated.

syms v1 m u1 rho_0 beta y1 s cd g Tmax delta_u delta_v delta_y
F = [v1+delta_v;(1/m)*(Tmax*u1+(rho_0/2)*exp(-beta*y1)*(v1^2)*s*cd+rho_0*exp(-beta*y1)*v1*s*cd*delta_v-(beta/2)*rho_0*exp(-beta*y1)*(v1^2)*s*cd*delta_y)-g+(Tmax/m)*delta_u];
% F = [v1+x(2);(1/m)*(Tmax*u1+(rho_0/2)*exp(-beta*y1)*(v1^2)*s*cd+rho_0*exp(-beta*y1)*v1*s*cd*X(2)-(beta/2)*rho_0*exp(-beta*y1)*(v1^2)*s*cd*X(1))-g+(Tmax/m)*delta_u]; (Expression for ode45)

[Torsten]

Whatever reason seems to hinder you to use ode45 - you will have to apply it for your problem.

[JXT119]

But ode45 only performs numerical integration, I need a symbolic solution.

[Torsten]

Sometimes the things we want are impossible to achieve.

Your ODEs are too complicated to allow a symbolic solution.

What by the way are the solution variables and what are given parameters in your F definition ?

y1 and v1 are solution variables and all others are fixed given parameters ?

dy1/dt = v1 + delta_v
dv1/dt = (1/m)*(Tmax*u1+(rho_0/2)*exp(-beta*y1)*(v1^2)*s*cd+rho_0*exp(-beta*y1)*v1*s*cd*delta_v-(beta/2)*rho_0*exp(-beta*y1)*(v1^2)*s*cd*delta_y)-g+(Tmax/m)*delta_u

?

[JXT119]

My solution variables are delta_v and delta_y.

y1 and v1 are nominal values around which I have linearized a non linear problem.

[Walter Roberson]

It is not clear to me what is to be integrated and what the variable of integration is and what the bounds of integration are.

syms v1 m u1 rho_0 beta y1 s cd g Tmax delta_u delta_v delta_y
F = [v1+delta_v;(1/m)*(Tmax*u1+(rho_0/2)*exp(-beta*y1)*(v1^2)*s*cd+rho_0*exp(-beta*y1)*v1*s*cd*delta_v-(beta/2)*rho_0*exp(-beta*y1)*(v1^2)*s*cd*delta_y)-g+(Tmax/m)*delta_u]

F = 

syms LB UB
int(F, delta_v, LB, UB)

ans = 

int(F, delta_y, LB, UB)

ans = 

Those look valid to me.

Answer 1

[Torsten]

So your equations are

d(delta_v)/dt = v1 + delta_v
d(delta_y)/dt = (1/m)*(Tmax*u1+(rho_0/2)*exp(-beta*y1)*(v1^2)*s*cd+rho_0*exp(-beta*y1)*v1*s*cd*delta_v-(beta/2)*rho_0*exp(-beta*y1)*(v1^2)*s*cd*delta_y)-g+(Tmax/m)*delta_u

with all parameters except delta_v and delta_y known values ?

Then the following approach should work. Look at your equations to see how a, b, c and d have to be set.

syms t x(t) y(t) 
syms a b c d real
eqn1 = diff(x,t) == a + x;
eqn2 = diff(y,t) == b*x + c*y + d

eqn2(t) = 

dsolve([eqn1,eqn2])

ans = struct with fields:
    y: exp(t)*(C1 + (a*b*exp(-t))/(c - 1)) + exp(c*t)*(C2 - (exp(-c*t)*(d - (d - a*b)/c))/(c - 1))
    x: -(exp(t)*(C1 + (a*b*exp(-t))/(c - 1))*(c - 1))/b