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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?

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    Your first equation has what is often called a *logarithmic derivative*; from the chain rule, we know that the derivative of $\log\,f(x)$ is $f^\prime(x)/f(x)$.2012-07-24
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    I'm also curious of this, and I know two Wiki articles that may help (though I did not have the patience to read it through) http://en.wikipedia.org/wiki/Differential_of_a_function#Other_approaches http://en.wikipedia.org/wiki/Differential_(infinitesimal)2012-07-27
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    I guess that we can do so, is guaranteed by the fundamental theorem of calculus involving Riemann Sums, showing that they are equivalent to the integral and that integration is inverse of differentiation.2012-07-28
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    Shouldn't this be three separate questions, rather than three-in-one?2012-08-02

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