Hello,

I try to solve this system of 2nd order differential equations in variables $x(t,k),y(t,k)$ and $z(t,k)$

$k^2y + 3 H \dot{y} = \frac{1}{2z_0} ( H \dot{z} - \dot{z} ) + \frac{\dot{z}_0}{2z_0} ( 2 H z + \dot{z} ), ~~~~~~~~ (1)$

$H x + \dot{y} =- \frac{a^2}{2 z_0} ( \dot{z}- H z ) + \frac{a^2 \dot{z}_0}{2 z_0} x, ~~~~~~~~ (2)$

$\left[\ddot{y} + H (2 \dot{y} + \dot{x}) + ( 2 \dot{H} + H^2 )x \right] \delta_{ij} + 1/2 ( \Delta \delta_{ij} - \partial_i \partial_j ) ( x-y) = \frac{a^2}{2z_0} ( \partial_i \partial_j z - H \dot{z} \delta_{ij}) ~~~~~~~~~ (3)$

Taking the trace of (3) yields:

$ \ddot{y} + H ( 2 \dot{y} + \dot{z} ) + ( 2 \dot{H} + H^2 ) x = - \frac{a^2}{2z_0} H \dot{z}, ~~~~~~~~ (3a) $

While the off-diagonal terms are:

$x= y- \frac{a^2 z}{2z_0} ~~~~~~~ (3b)$

And the last equation

$\ddot{z} - H \dot{z} + 3k^2 z + 2 (x-3y) \ddot{z}_0 - 2 H (x-3y) z_0 + 2 (\dot{x} - 3 \dot{y} ) \dot{z}_0 =0 ~~~~~~~~~~(4)$

where ( $\dot{z}$) means the derivative with respect to time. $H(t), a(t), z0((t)$ are given functions as in the attached MA file

Now since I try to get three unknown functions, one should to solve only three equations to prevent redundant . So I wonder which equation should be used with (1) and (2), is it Equ. (4)? cause Equ. (3) has two parts?

Anyway here is my procedure in solving so if any help to know whether it's correct or no and what is the correct procedure.

I used the redundancy of the equations to substitute by the higher order derivatives as following and as in the MA code:

• Substitute by $x$ from (3b) in (2) and (3a)
• Substitute by $\dot{y}$ from (2) in (1) and (3)
• Substitute by $\ddot{z}$ from (4) in (1) and (3)
• Now I get a couple of equations in $(\ddot(y), y, z, \dot{z})$, which could be solved by AsymptoticDSolveValue

Any help is appreciated!