I have an ODE of the form y''(x)=F(x,y') which has the initial conditions, $y(\mu)=\mu$ and y'(\mu)=1. Now I have seen that an equation which is equidimensional in x, can be made autonomous by the change of variable $x=e^t$, ie now we must solve an ODE for the unknown $z(t)$ ($=y(x)$). But now what happens to the initial conditions??
I often see change of variables of some sort in books. But no one really talks about what happens to the initial conditions.
I guess people dont need to transform the initial conditions, since if a geneneral solution $z(t)$ to the transformed (and simplified) equation can be found. Then we can resolve any arbitrary constants, using the fact $z(ln(t))=y(x)$. But I'm trying to solve my equation through power series techniques so I dont think this method applies. I need the conditions in advance to compute the coefficients of the terms in the power series.
I think the chain rule may be applied to evaluate one of the conditions. That is by application of the chain rule we have $\frac{dy}{dx}=\frac{dy}{dt}\frac{1}{t}$, so $\frac{dy}{dt}(ln(\mu))=\mu$ since $\frac{dy}{dx}(\mu)=1$. But what about the first condition? how is that transformed?