Differential equation with functions as coefficients Do you have any ideas about how to solve a differential equation of the following form, without using series solutions? 
$$f(x)y''(x) + g(x)y'(x) + h(x)y(x) = 0$$
Edit: I'd like to solve something along the lines of: 
$$y''(x)+\frac{1}{x}y'(x)+\frac{t}{x^2}y(x) = 0$$ where $t$ is a parameter.
 A: $$f(x)y''(x) + g(x)y'(x) + h(x)y(x) = 0 \tag 1$$
It is a second order homogeneous linear ODE. With the change of function :
$$Y(x)=\exp\left(\frac{1}{2}\int \frac{g(x)}{f(x)}dx\right)y(x),$$
the equation $(1)$ is transformed to :
$$\frac{Y''(x)}{Y(x)}=\Psi(x) \tag 2$$
where $\Psi(x)$ is a function expressed with $f(x),g(x),h(x)$ into it.
Eq.$(2)$ is also a second order homogeneous linear ODE, but simpler to classify  than $(1)$ : Only one function $\Psi(x)$ is involved instead of three functions in $(1)$.
The question is : Can we analytically solve $(2)$ ?
One cannot answer on a general manner. The results can be very divers depending on the kind of function $\Psi(x)$. For examples ( non exhaustive of course) :
$\Psi(x)=a \quad\to\quad Y(x)$ involves $\exp$ , $\cosh$ , $\sinh$ , $\cos$, $\sin$ functions.
$\Psi(x)=ax+b \quad\to\quad Y(x)$ involves Airy functions.
$\Psi(x)=ax^2+bx+c \quad\to\quad Y(x)$ involves Parabolic Cylinder functions.
$\Psi(x)=ax^b \quad\to\quad Y(x)$ involves Bessel functions.
$\Psi(x)=ae^{bx}+c \quad\to\quad Y(x)$ involves Bessel functions.
$\Psi(x)=\frac{a}{x+b} \quad\to\quad Y(x)$ involves Bessel functions.
$\Psi(x)=\frac{a}{x^2}+\frac{b}{x}+c \quad\to\quad Y(x)$ involves Confluent Hypergeometric and/or Kummer and/or Whittaker functions.
And so on...
Many others involves Hypergeometric functions to express $Y(x)$ on closed form.
But more generally, in many cases one cannot express the solution on closed form because a convenient special function is not available.
This explain why your question is too large to be answered on a general manner. You have to restrict your question to particular forms of ODE if you expect an answer about a specific method to solve it.
