I am trying to solve the differential equation
$ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $
where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} $.
for $1\ll{x}$, the equation simplifies to
$ \frac{d^{2}f(x)}{dx^{2}}+\frac{2}{x}\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $
Substituting, $k x^{p}$ for $f(x)$, and solving for $p$ and $k$ gives the solution as $f(x)=c(x^{4/3})$, where $c$ is some constant.
for $x\ll{1}$, the equation simplifies to
$ x\frac{d^{2}f(x)}{dx^{2}}+\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $
Substituting, $k x^{p}$ for $f(x)$, and solving for $p$ and $k$ gives the solution as $f(x)=c(x^{2/3})$, where $c$ is some constant.
However, I could only solve the equation for $x\ll{1}$ and $1\ll{x}$, Is it possible to solve this equation for all $x$?