Suppose that $u(x)$ satisfies the equation

$$\frac{d^{2} u}{d x^{2}}-f(x) u=0$$

where $f(x)$ is a given non-zero function. Show that under the change of coordinates $x=x(t)$,

$$\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0$$

where a dot denotes differentiation with respect to $t$. Furthermore, show that the function

$$U(t)=\dot{x}^{-\frac{1}{2}} u(x)$$

satisfies

$$\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0$$

Choosing $\dot{x}=(f(x))^{-\frac{1}{2}}$, deduce that

$$\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0$$

for some appropriate function $F(t)$. Assuming that $F$ may be neglected, deduce that $u(x)$ can be approximated by

$$u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),$$

where $c_{+}, c_{-}$are constants and $A, G$ are functions that you should determine in terms of $f(x)$.