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What you are suggesting is fine. In general, you can always define $C_3=f(C_1,C_2)$ as long as doing so eliminates either $C_1$ or $C_2$ from the equation. While it is still technically correct otherwise, you wind up with three constants which are then no longer arbitrary, which only serves to make the solution more confusing.


The general solution of a linear ODE can be written as $$ y=y_{h}+y{c} $$ where $y_{h}$ stands for the homogeneous solution and $y_{c}$ is the complementary particular solution. In your case, the ODE $$ y''-4y'+4y=0 $$ has homogeneous solutions $c_{1}e^{2x}$ and $c_{2}xe^{2x}$. You can easily verify that $x^2e^{2x}+x-2$ is particular solution for $$ ...


The golden ratio comes up in a nice way when one is looking at the rate of convergence of the Secant Method of root finding. In the Newton-Raphson Method, in good situations, the number of correct decimal places roughly doubles with each iteration. With the Secant Method, the number of correct decimal places, in good situations, gets roughly multiplied by ...


There are a whole bunch of ways to determine the values of different constants. I'll tell you about some of the historic methods. $\pi$ One of the oldest methods used was the exhaustion method, by Archimedes. He put a circle in between two polygons of equivalent sides and gradually increased the number of sides until the two became nearly identical, ...


Unit vectors in the opposite direction along the real line in the complex plane add to zero.


There is a problem with your definition of $\infty$. You can't say that $\infty$ is the number larger than any finite number, because $\infty$ is not a number in the first place. You can't say that some set of numbers ($\mathbb N$ or $\mathbb R$, for example) contains $\infty$ if you want usual rules to hold; for example from $$\infty + a = \infty$$ ...


I once chanced upon this book at the library: A Dictionary of Real Numbers. It's a very interesting endeavour, to say the least. And maybe you'll find it similar to what you're looking for.


As pointed out in the comments, the OEIS is probably your best bet. It contains nearly 10,000 constant sequences, which you can look up by typing in their decimal expansion: OEIS search for 3.14159 RIES is a good resource for numbers constructed as simple formulas. Plouffe has a private version of the ISC which is too large to be available online, but ...

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