# Problem solving 2nd order ODE not for the faint of heart

## Main Question or Discussion Point

Problem solving 2nd order ODE...not for the faint of heart!!

Hey folks,

I'm having problems solving the following set of ODE's:

$$3H_a^2+H_b^2+6H_aH_b=k_1\rho$$ eq.1

$$\dot{H_a}+3H_a^2+2H_aH_b=k_2\rho$$ eq.2

$$\dot{H_b}+2H_b^2+3H_aH_b=k_3\rho$$ eq.3

These are cosmological equations. Note, $\rho=\frac{1}{b^6}(1-b^2+b^4)$, also the H's are Hubbles constant in a and b, eg

$$H_a=\frac{\dot{a}}{a}$$

$$H_b=\frac{\dot{b}}{b}$$

The a's and b's are functions of t (time) and the k's on the RHS are just constants. I want to put this all together and ultimately plot a as a function of t and b as a function of t.

The equations originate from the paper: http://arxiv.org/abs/0707.1062 , equations 9,10,11 and I am trying to duplicate the plots in fig1.

What I'm thinking:

Solve eqtn 1 for [itex]H_a[H_a] using the quadratic eqtn then plug that into 3 and use DSOLVE in mathematica.

Can anyone let me know if this is the correct approach.

Richard

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