Integrating two ODE'S

I'm trying to integrate these two equations: d/da(rho*a^3) = -3*w*rho*a^2; ((da/dt)/a)^2 + k/a^2 = 8*pi*G/3*rho

where a==a(t), rho==rho(a),w=w(rho).

For now, I am taking 'w' to b a constant. I found out d(rho)/dt = d(rho)/da * da/dt to get 'ode1' and expressed rhoas a function of time 't'. 'ode2' is just the second equation.

Code:

---------------------

syms a(t) rho(t) w(rho)

G=1;

w=1/3;

k=0;

ode1 = diff(rho) == -3*rho*(w+1)*sqrt(8*pi*G/3*rho);

ode2 = (diff(a)/a)^2 + k/a^2 == 8*pi*G/3*rho;

odes = [ode1;ode2];

Sol = dsolve(odes)

aSol(t) = Sol.a;

rhoSol(t) = Sol.rho

---------------------------- Output:

Sol =

struct with fields:

      a: [1×1 sym]
    rho: [1×1 sym]

aSol(t) =

C5

rhoSol(t) =

0

K>>

This is of course not the solution. Can someone please tell me where I'm going wrong. Thanks.

I am open to numerically integrating these as well

[David Goodmanson]

Hello Ian, For w = constant, since rho = rho(a) only depends on time implicitly through a(t), it appears that the solution to the first equation is simply

rho = C*a^(-3(w+1))

and then a(t) can be calculated from the second equation.

[Ian DSouza]

Thanks David. Yes,that is the solution for w=constant. I decided to keep the ODE because I wanted to eventually vary 'w' as a function of rho. But I got it to work using ODE45 now.