Not sure if I should be asking this in maths or physics section so please excuse me if asked in the wrong section. I am trying to numerically integrate the equations obtained in the exponential potential scalar field model of dark energy. The set of equations I have obtained is:
\begin{align}
x &= tH_{0} \\
y_{1} &= \frac{a}{a_{0}} \\
y_{2} &= \frac{\psi}{\psi _{0}}\\
y_{3} &= \frac{dy_{2}}{dx}
\end{align}
which on differentiating w.r.t x I get:
\begin{align}
{y_{1}}\prime &= y_{1}\left(\frac{\Omega_{m}}{y_{1}^3} + \frac{y_{3}^2}{6.0} + \frac{V(y_{2})}{2.H_{0}^2}\right) \\
y_{2}\prime &= y_{3} \\
y_{3}\prime &= -3\frac{y_{1}\prime}{y_{1}}y_{3} - \frac{1}{H_{0}^2}\frac{\partial v(y2)}{\partial y_{2}}
\end{align}
My doubt here is I am using RK4 method for solving this and when I write the increment should I increment the independent variable $x$ too?Like when I write formulae for $k_1,\,k_2,\,k_3,\,k_4$ etc I am incrementing $y_1,\,y_2,\,y_3$. As far as I know it is not needed as these ODE are autonomous equations yet when I try to plot them I am not getting the expected result.
Another doubt is I am plotting it w.r.t scale factor( $a=0$ to $a=1$) and when I plot $y_1$ which is $y_{1} = \frac{a}{a_{0}}$ shouldn't I be getting a linear curve or am I missing on some major concept?