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 Jun3 comment Inverting the sum of a Dense and Diagonal matrix @Eric, I went through the same set of trials with my research, so batting zero is average! I'd recommend the books by G. W. Stewart (1, 2). Eventually, there will be 5 volumes, but the first two alone will improve how you think about matrix problems. Jun3 revised Inverting the sum of a Dense and Diagonal matrix added response to second part Jun2 revised Inverting the sum of a Dense and Diagonal matrix added link to review article Jun2 answered Inverting the sum of a Dense and Diagonal matrix Jun1 comment Integration of a first order ODE just from looking at it, it is obvious that $y(t)=ce^{5t}$ is a solution. But, what I've always struggled with is, is there a systematic way of choosing the particular solution? My issue is it seems odd that we can systematically, for the most part, find the homogenous solution, yet in finding the particular solution, we are reduced to guessing. Yes, they're educated guesses, but there is still an element trial and error that is bothersome to me. May24 answered A series solution of the differential equation: $\frac{d^2u(x)}{dx^2}+ u(x)^n = 0.$ Apr20 revised What is the signature of a matrix? fixed notation Apr20 suggested approved edit on What is the signature of a matrix? Apr19 awarded Enthusiast Apr18 comment On maximizing a function A few questions: is $n$ real or is it an integer? what do you know about $b$? what about $q_i$? what is the range on your sum? The reason I ask, is that if your sum range is small and n is an integer (with a small $N$), then if may be easiest just to calculate every possible value. Apr18 comment Find the spectrum of the linear operator $T: \ell^2 \to \ell^2$ defined by $Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$ question on notation, does your statement $\theta \in ]0,1[$ mean that the set is non-inclusive? Apr18 answered Why this spectrum problem is self-adjoint? Apr18 comment Why this spectrum problem is self-adjoint? This has me confused, and I believe your notation is a hindrance. Let the dagger, $\dagger$, denote the conjugate-transpose, or adjoint, i.e. $A^\dagger = (A^\ast)^T$, where $\ast$ denotes complex conjugation. Now, if $M$ is Hermitian (or, self-adjoint), then $M^\dagger = M$. This implies that the diagonal of $M$ must be real, so $k = i c$ where $c \in \mathbb{R}$. Then your eigenvalues are $\pm \sqrt{c^2 + |q(x)|^2}$. So, with the notation changes could you clarify why your confused? Apr16 revised Parallel Projection Of An Ellipse formatted equations Apr16 suggested approved edit on Parallel Projection Of An Ellipse Apr12 accepted Under what conditions is integrating over a series expansion valid for an improper integral? Apr12 revised Under what conditions is integrating over a series expansion valid for an improper integral? reverted Apr10 comment Under what conditions is integrating over a series expansion valid for an improper integral? We determined that the pole at the origin was the cause of the problem. But, I also found that if the integration limits each corresponded to a pole, i.e. $x \in [\pi(n-1), \pi n]$, you are unable to compensate for the pole. But, if you shifted the integral to $x \in [\pi(n - 1/2), \pi (n + 1/2)]$ the poles can be compensated for, and the series of integrals is an alternating series, $i \sum (-1/e)^n$, which converges to $-i/(1-e)$. Apr10 revised Under what conditions is integrating over a series expansion valid for an improper integral? added addt'l request from the community Apr10 comment Under what conditions is integrating over a series expansion valid for an improper integral? @JM, I have no idea, but it did bring up some questions on how to proceed.