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What are the various physical mechanisms for energy transfer to the photon during blackbody emission? Given a recurrence relation:   $$ x_{n+2} = px_{n+1} + qx_n + r $$   Find the general term $x_n$ given initial conditions $x_1 = a$ and $x_2 = b$, where $a,b,p,q,r$ are some given numbers, $n \in \mathbb N$.   Since this is a non-homogeneous linear recurrence relation i've started with solving for $x_n^h$ which is a solution for homogeneous relation:  $$ x_{n+2} - px_{n+1} - qx_n = 0 \\ \lambda^2-p\lambda - q = 0 $$  I'm assuming here that two distinct roots $\lambda_{1,2}$ exist for this equation (single root case is handled similarly). So the general term is in the form:  $$ x_n = C_1\lambda_1^{n-1}+C_2\lambda_2^{n-1} $$  With the initial condition it's possible to find the values for $C_1$ and $C_2$:  $$ C_1 + C_2 = a \\ C_1\lambda_1 + C_2\lambda_2 = b $$  After solving the system of equation with respect to $C_1$ and $C_2$:  $$ C_1 = \frac{b-a\lambda_2}{\lambda_1 - \lambda_2} \\ C_2 = \frac{b-a\lambda_1}{\lambda_2 - \lambda_1} \\ $$  Which eventually results in:  $$ x_n^h = \frac{(a\lambda_2 - b)\lambda_1^{n-1} - (a\lambda_1-b)\lambda_2^{n-1}}{\lambda_2 - \lambda_1} $$  Final solution consists of a sum of particular and homogeneous solutions, but that is where I got stuck. What is the way to find the particular solution for the given problem? I'm pretty new to such kind of problems so any details are very much appreciated.  upd:  Based on the technique in @rtybase answer here are my findings:  $$ \begin{align} x_1 &amp;= a \\ x_2 &amp;= b \\ x_3 &amp;= pb - qa + r \end{align} $$  Writing the system for $c_1, c_2, c_3$:  $$ \begin{align} a &amp;= c_1 + c_2 + c_3 \\ b &amp;= c_1\lambda_1 + c_2\lambda_2 + c_3 \\ pb+qa+r &amp;= c_1\lambda_1^2 + c_2\lambda_2^2 + c_3 \end{align} $$  Now solving for the coefficients gives a huge formula. The answer to this problem suggests that:  $$ x_n = \frac{(\lambda_2(a+\gamma) - b -\gamma)\lambda_1^{n-1} - (\lambda_1(a + \gamma) - (b-\gamma))\lambda_2^{n-1}}{\lambda_2 - \lambda_1} - \gamma $$  Where:  $$ \gamma = \frac{r}{p+q-1} $$  Unfortunately I was not able to arrive at exactly the same result, but testing my solution (not posting since it's monstrously big) using SameQ in Mathematica with the given answer shows that they are equivalent. Those transformations are not very obvious to me, so i skipped them since the answer matches the given one (yet the symbolic expressions are different) How can I echo a newline in a batch file? How can you you insert a newline from your batch file output?  I want to do something like:  echo hello\nworld   Which would output:  hello world Frequency count of two column in R Selecting text in profile activity comments causes unexpected clipping I read recently that if you hold a bullet in one hand and a pistol in the other, both hands at the same height, and subsequently fired the pistol at the same time as dropping the bullet, both bullets would hit the ground at the same time (assuming the fired bullet encounters no obstacles).  Is this true, and if so can you explain the reasoning behind it with as few equations as possible? I assume it is essentially that gravity is unaffected by momentum, but it just sounds counter-intuitive. When I run LaTeX file its gives error as  "! Package minted Error: You must have pygmentize installed to use this package", you must have pygments installed. when I go through minted documentation minted itself required pygments, so Tex friend How do I install this pygment, I don't know the procedure to install. Please guid me or any link pygement.exe file There is a command l available on my machine which appears to do nothing.  which l also produces no output. Is this a real command, and does it actually do anything? Best Place to put Feature Requests? The only place I am aware of is the . What is the use for non-inverting input resistance in a negative feedback opamp? Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? ClassNotFoundException: org.eclipse.jetty.util.component.AbstractLifeCycle running inside jetty Which prepositions are followed by accusative pronouns instead of reflexive pronouns? I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? I received $1000 and was asked to send it back. How was this scam meant to work? Possible Australianism - netbank = a website + app by which an Aussie can do their banking, including transferring money to other people.  A while ago, I ended up with $1000 transferred into my account, which in the description looked like a netbank transfer, not like a cheque, cash deposit etc, and which looked like it had already cleared and landed in my account. Usually when something is still working through the bank's system it displays as 'pending', but did not in this case.  I received a phone call from someone, saying that they had accidentally typed in the wrong account number, and wanting me to send them the same amount of money back via netbank.   Surely bank account numbers have a checksum, which make it relatively difficult for a typo to result in a payment going to the wrong person? What are likely sources of them being able to find my phone number to call me? Obviously, the bank either being hacked, or giving it out in response to a query from this person (but surely the bank is unlikely to tell me my own mobile number if I claim to have forgotten it?). Presumably either my phone, or the phone of someone I have sent money to/received money from could be compromised. Has anyone heard of any instances of this happening?  I then went to my bank, explained the situation and said that I did want to refund the money through the officially sanctioned methodology. After the laughter died down, the bank told me to ask the person to fill out a form, then they would get me to fill out a form that would authorize undoing that exact payment.  I rang up the person to tell them this, and over the course of about a week received an increasingly elaborate story about how that wasn't working, they needed the money immediately, there was a fee to fill out that form and they couldn't afford to pay it etc.  After that, I never heard back from them again. I check my netbank for several weeks after, expecting the money to magically disappear, but it did not. This was probably a year ago, and I'm pretty vague with money so may not have noticed if it disappeared more recently. How were they expecting to benefit?  Were they hoping to get the transfer from me, and undo the transfer to me, to leave them $1k up? If so, how can an internet banking transfer be 'undone' once it has already shown as 'received'. Or was this to do with money laundering? The account they requested me to send money to was the same as the one that the money appeared to come from. This was a scam, right? sudo: /etc/sudoers is owned by uid 501, should be 0 To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). How to deploy EJB based application on Tomcat Track next Socratic badge progress Just got the  on StackOverflow and, since it can be awarded multiple times, wanted to start tracking progress to the next one, but, the badge is not shown in Available badges to track anymore:    Is it possible to track already awarded badges that can be awarded multiple times? Does "Going Infinite" necessarily end the game? I am trying to recall the name of a story I read when I was fairly young (maybe 20 years ago, but not sure how old the story is).    What I remember: A man acquires a set of creatures that he keeps in his house, I think in an aquarium.  They evolve over time, quickly, and think of him as their benevolent god. They build statues in his likeness.   Then, I believe some sort of accident happens, and the statues they build become a little less benevolent, which bothers him.  This pattern continues on with his cruelty increasing and their depiction of him becoming more cruel. I'm giving training to a large group of people remotely and will be covering a number of application shortcut keys.  I've seen 's on-screen display of keystrokes; is there a similar product for Windows?    The other features of MousePose (fancy mouse pointers, etc.) aren't really useful to me, just the OSD of key presses. I was wondering if mobs can go through doors or not. It's the difference between me just putting up a door to my house, or putting up dirt instead.