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I'm doing a homework problem, but I'm not sure on whether I'm doing the change of variable correctly for the ODE. We are given the ODE of $y'' + e^{2x}y= 0$, and asked to make the change of variable $y(x) = z(\epsilon)$, with $\epsilon = e^x$. Here's what I did so far:

$$\frac{dy}{dx} = \frac{dy}{d\epsilon}\frac{d\epsilon}{dx} = \frac{dz}{d\epsilon}e^x$$ $$\frac{d^2y}{dx} = \frac{d}{dx}\bigg(\frac{dy}{dx}\bigg) = \frac{d}{dx}\bigg(\frac{dz}{d\epsilon}e^x\bigg) = \frac{d}{dx}\bigg(\frac{dz}{d\epsilon}\bigg)e^x + \frac{dz}{d\epsilon}e^x = \frac{d^2z}{d\epsilon^2}e^{2x}+\frac{dz}{d\epsilon}e^x$$ However, subsituting that into the ODE gives us $e^{2x}z''+e^xz'+e^{2x}z=0$, but I'm kind of confused on where to go from there. Did I do something in my change of variables wrong, or is there just some way to solve this ODE that I'm missing?

EDIT: Whoops put a z instead of an x somewhere in the work, but it still has the same result...

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Whoops, I realized that since $\epsilon = e^x$, I should substitute that in so I get $\epsilon^2z''+\epsilon z'+\epsilon^2z=0$, which I can solve since it is a Bessel Equation.

I got $y(x) = c_1J_0(e^x)+c_2Y_0(e^x)$ for those curious.

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