Methods to solve differential equations We are given the equation
$$\frac{1}{f(x)} \cdot \frac{d\left(f(x)\right)}{dx} = x^3.$$ 
To solve it, "multiply by $dx$" and integrate:
$\frac{x^4}{4} + C = \ln \left( f(x) \right)$ 
But $dx$ is not a number, what does it mean when I multiply by $dx$, what am I doing, why does it work, and how can I solve it without multiplying by $dx$?  
Second question:
Suppose we have the equation $$\frac{d^2f(x)}{dx^2}=(x^2-1)f(x)$$
Then for large $x$, we have $\frac{d^2f(x)}{dx^2}\approx x^2f(x)$, with the approximate solution $ke \cdot \exp \left (\frac{x^2}{2} \right)$
Why is it then reasonable to suspect, or assume, that the solution to the original equation, will be of the form $f(x)=e^{x^2/2} \cdot g(x)$, where $g(x)$ has a simpler form then $f(x)$? When does it not work?  
Third question:
The method of replacing all occurences of $f(x)$, and its derivatives by a power series, $\sum a_nx^n$, for which equations does this work or lead to a simpler equation?
Do we lose any solutions this way?

First question only: The differential equation can be rewritten as 
$$\frac{f'(x)}{f(x)}=x^3.$$
Find antiderivatives of both sides.  To calculate $\int \frac{f'(x)}{f(x)}\,dx$, note that by the Chain Rule, $\frac{f'(x)}{f(x)}$ is the derivative of $\log|f(x)|$.
We conclude that $\log|f(x)|+C_1=\frac{x^4}{4}+C_2$ for some constants $C_1$ and $C_2$, which may be combined into one.
There was no multiplication by $dx$. Yet we get the same result as by the admittedly dubious procedure of multiplying by $dx$.  So at least the dubious procedure is harmless, it gives the correct result. So it can be thought of as a mnemonic device that helps us get to the right answer. 
Remark: There is more to it than that, the process captures a certain useful physical intuition. And one can even make mathematical sense of naked $dx$. 
