I have the following system of ordinary differential equations up to fourth order (one of the equations is like a restriction for the other) for the one variable real functions $a(t)$ and $b(t)$:
$0=\frac{-12 \gamma (3 \alpha -\beta ) a(t)^2 b(t)^2 a''(t)^2-24 \gamma (3 \alpha -\beta ) a(t)^2 b(t) a'(t)^2 b''(t)+60 \gamma (3 \alpha -\beta ) a(t)^2 a'(t)^2 b'(t)^2-36 \gamma (3 \alpha -\beta ) b(t)^2 a'(t)^4+6 \gamma a(t)^2 b(t)^4 a'(t)^2+24 \gamma (3 \alpha -\beta ) a^{(3)}(t) a(t)^2 b(t)^2 a'(t)-24 b'(t) \left(\gamma (3 \alpha -\beta ) a(t) b(t) a'(t)^3+2 \gamma (3 \alpha -\beta ) a(t)^2 b(t) a'(t) a''(t)\right)+24 \gamma (3 \alpha -\beta ) a(t) b(t)^2 a'(t)^2 a''(t)-\Lambda a(t)^4 b(t)^2}{2 \gamma a(t)^4 b(t)^4}$
$0=-\frac{8 \gamma (3 \alpha -\beta ) a(t)^3 a^{(4)}(t) b(t)^3+12 \gamma (3 \alpha -\beta ) a(t)^2 b(t)^3 a''(t)^2-8 \gamma (3 \alpha -\beta ) a(t)^3 b^{(3)}(t) b(t)^2 a'(t)-120 \gamma (3 \alpha -\beta ) a(t)^3 a'(t) b'(t)^3+12 \gamma (3 \alpha -\beta ) b(t)^3 a'(t)^4+2 \gamma a(t)^2 b(t)^5 a'(t)^2+16 \gamma (3 \alpha -\beta ) a(t)^2 a^{(3)}(t) b(t)^3 a'(t)+60 b'(t)^2 \left(2 \gamma (3 \alpha -\beta ) a(t)^3 b(t) a''(t)+\gamma (3 \alpha -\beta ) a(t)^2 b(t) a'(t)^2\right)-16 b''(t) \left(2 \gamma (3 \alpha -\beta ) a(t)^3 b(t)^2 a''(t)-5 \gamma (3 \alpha -\beta ) a(t)^3 b(t) a'(t) b'(t)+\gamma (3 \alpha -\beta ) a(t)^2 b(t)^2 a'(t)^2\right)+4 a''(t) \left(\gamma a(t)^3 b(t)^5-12 \gamma (3 \alpha -\beta ) a(t) b(t)^3 a'(t)^2\right)-4 b'(t) \left(12 \gamma (3 \alpha -\beta ) a(t)^3 a^{(3)}(t) b(t)^2-12 \gamma (3 \alpha -\beta ) a(t) b(t)^2 a'(t)^3+\gamma a(t)^3 b(t)^4 a'(t)+18 \gamma (3 \alpha -\beta ) a(t)^2 b(t)^2 a'(t) a''(t)\right)-\Lambda b(t)^7}{2 \gamma a(t)^2 b(t)^7}$
Where I am using prime to denote derivative with respect to time $t$.
Here, $\alpha$,$\beta$,$\gamma$ and $\Lambda$ are constants of $O(1)$. Of course, a lot of initial conditions need to be given. Some reasonable initial conditions from the context of the problem are $a(t=1)=b(t=1)=1$, and their first derivatives should be positive $a'(t=1),b'(t=1)>0$ but not so big (0.5, for example). I don't have more information in order to give solid initial conditions for the higher derivatives, but it is reasonable again to choose them positive and the initial condition for the second derivative $b''(t=1)$ can be obtained from the first equation by consistency. One could choose, for example: $ a(1) = 1, a'(1) = 1, a''(1) = 1/2, a'''(1) =1/2 , b(1) = 1, b'(1) = 1, b''(1) = -(103/24) $
I would like to know if it's possible to know something about the approximate solution of this system, for example expanding $a(t)=\sum_k^n a_kt^k$ and $b(t)=\sum_k^n b_kt^k$.