Solving a system of differential equations, one second order and one first order equation

Hello,

I am trying to solve a system of differential equation, but one of them is first order, and the other secod order:

% d2x = -x -(sigma_0*z+sigma_1*exp(-(dx/v_d)^2)*dz+sigma_2*dx);

% dz = dx-sigma_0*abs(dx)*z/(f_c+(f_s-f_c)*exp(-(dx/v_s)^2));

where sigma_0, sigma_1, sigma_2, v_d, v_s, f_c,f_s are constants.

I tried to solve them using ode45, as i have done to solve 2 second order diffecrential equation before so:

y1=x, y2=dx, y3=z, y4=dz

dy1=y2, dy2=-y1 -(sigma_0*y3+sigma_1*exp(-(y2/v_d)^2)*y4+sigma_2*y2)

dy3=y4, dy4=? I dont have d2z, since its a first order diferential eq so I am not sure how to proced here or if there is another way to solve it.

Thank you!

Answer 1

[Sam Chak]

Hi @Angie

If the ODEs are

then they can be rewritten as

.

Solving the systems of equations

.

Now, you can enter ODEs as follow:

.

[Angie]

Hi @Sam Chak,

Thank you for your answer! Indeed, I can replace the dz in the rhs of my first equation d2x with the expression of dz from my second equation. Then my first equation d2x will be in terms of x,dx and z only and I can write its state space representation to obtain two first order equations; dy1 = dx and dy2 = f(x,dx,z). Meanwhile dy3 = dz = g(x,dx,z).

Thanks again!