User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

DiffEQ Ongoing Project
Assignment:


 * $\displaystyle \int_{\to 0}^{\to 1} \ln x \ln \left({1-x}\right)\ \mathrm dx$

To prove the integral exists, note that $\ln$ is continuous for all $x$ in its domain.

As $x \to 0^+$:

As $x \to 1^{-}$:

Find the power series of $\ln(1-x)$:

Then,

Let $n \in \N_{\ge 1}$

So,

Since $n$ was arbitrary, this holds for all cases $n \ge 1$, for $x \in (0\,.\,.\,1)$.

--GFauxPas (talk) 13:41, 25 March 2014 (UTC)

Question, Proofwiki'ers: What's the easiest sufficient condition to use $\int \sum = \sum \int$ in what I did? --GFauxPas (talk) 13:00, 26 March 2014 (UTC)

Second Order Linear Homogeneous Diff Eq
Let $A, B, C$ be constants.

Consider the second order linear homogeneous diffeq:


 * $A\dfrac {\mathrm d^2y}{\mathrm dx^2} + B \dfrac {\mathrm dy}{\mathrm dx} + C y = 0$

The characteristic equation of this is:


 * $Ar^2 + Br + C = 0$

where $y = e^{rx}$

--GFauxPas (talk) 17:04, 26 March 2014 (UTC)