So I have tried many methods including Laplace transform and differentiation but the p and q seems to be unsolvable, I would like to know how does the form change from the first equation, then substitude the Y = e^t to get the second equation. also my second question is why is how can i solve for p and q if i dont have any initial conditions?
If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation:  $$z^2w'' + \alpha zw' + \beta w = 0$$  where $w$ is a function of $z$ and $\alpha, \beta$ are constants. How would the change of variables $t =\ln(z)$ transform this equation?
It is quite straightforward to find the fundamental solutions for a given Pell's equation when $d$ is small. But I am unable to solve this equation, as I'm unable to find the fundamental solutions:  Solve: $x^2-29y^2=1$ and $x^2-29y^2=-1$ with $y\not=0$.  Could you please guide me through the solution Can I use Bluray as a boot cd? PCA on correlation or covariance? What are the main differences between performing principal component analysis (PCA) on the correlation matrix and on the covariance matrix? Do they give the same results? How to create a Windows 7 installation USB from OS X? I'm trying to create a Windows 7 installation USB (for use on a PC) from OS X.  I've tried following these instructions from a :   Insert the USB flash drive and run the command diskutil list to find out the disk name, we'll use /dev/disk1 as an example Now unmount the disk using diskutil unmountDisk /dev/disk1 dd if=/path/to/Win7.iso of=/dev/disk1 bs=8192   But the instructions don't work. When I insert the created USB drive into my PC and try to boot from it, I get this message:     Reboot and Select proper Boot device or Insert Boot Media in selected Boot device and press a key   What am I missing? Story where people inject "books" in the back of their necks? I read this story in middle school or high school and I cannot for the life of me remember the name of it. I think it started off in a small clan of people, there was a young boy who had just come of age and he had some kind of gift that related to the books? Also I think the injecting was illegal, because I vaguely remember the boy encountering an old man who had a bunch of books that he had to keep hidden. There was also a city and he met a girl there that was Special some how. I may just be merging memories but I think at some point the boy and girl went to an underground city?  Does anyone know what I'm talking about? Consider the following 2-variable function:     $$f(x,y) = \frac{x^2y}{x^4+y^2}$$   I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$.   I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, and it went as the following:  $$ x = r \cos \theta, \qquad y = r\sin\theta $$   Hence,  $$\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^4 + y^2} = \lim_{r \to 0}\frac{r^2\cos^2\theta(r\sin\theta)}{r^4\cos^4\theta + r^2\sin^2\theta}$$  This simplifies to,  $$ \lim_{r \to 0} \frac{r^3 \cos^2\theta\sin\theta}{r^2(r^2\cos^4\theta + \sin^2\theta)}$$  Simplifying $r^3/r^2$, we finally get;  $$\lim_{r \to 0} \frac{r (\cos^2\theta\sin\theta)}{r^2\cos^4\theta + \sin^2\theta}$$  Now from the above, we find that as $r \to 0$ the limit is $0$.   I wanted to verify this answer so I checked on Wolfram Alpha. Explicitly without changing to polar coordinates, it said that the limit does not exist at $(0,0)$ and rightly so. Then how is it that with polar coordinates, the limit exists and is $0$? Am I doing something wrong in this method?  Also, what should I do in this situation, and when should I NOT use polar coordinates to find limits of multi-variable functions? Interfaces with static fields in java for sharing 'constants' I'm looking at some open source Java projects to get into Java and notice a lot of them have some sort of 'constants' interface.  For instance,  has an interface called , and most other core classes implement this interface. The interface is riddled with static members. Is there a reason for this approach, or is this considered bad practice? Why not use enums where it makes sense, or a static class?  I find it strange to use an interface to allow for some sort of pseudo 'global variables'.  public interface PConstants {    // LOTS OF static fields...    static public final int SHINE = 31;    // emissive (by default kept black)   static public final int ER = 32;   static public final int EG = 33;   static public final int EB = 34;    // has this vertex been lit yet   static public final int BEEN_LIT = 35;    static public final int VERTEX_FIELD_COUNT = 36;     // renderers known to processing.core    static final String P2D    = "processing.core.PGraphics2D";   static final String P3D    = "processing.core.PGraphics3D";   static final String JAVA2D = "processing.core.PGraphicsJava2D";   static final String OPENGL = "processing.opengl.PGraphicsOpenGL";   static final String PDF    = "processing.pdf.PGraphicsPDF";   static final String DXF    = "processing.dxf.RawDXF";     // platform IDs for PApplet.platform    static final int OTHER   = 0;   static final int WINDOWS = 1;   static final int MACOSX  = 2;   static final int LINUX   = 3;    static final String[] platformNames = {     "other", "windows", "macosx", "linux"   };    // and on and on  } Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above? How are the Tenchi timelines related? Between Muyo, Universe, GXP, and Ryo-ohki, I'm thoroughly confused. Sometimes Tenchi series seem to be related (Ryo-ohki and Universe) and sometimes not (GXP). What are all the Tenchi series, are they related, and if they are, how? Give us better search-by-score syntax Picking random points in the volume of sphere with uniform probability My eth0 has gone and I don't have internet and network connection I had internet connection, but when I upgraded my VGA and restarted my PC then everything for eth0 has gone.     When I used ifdown:     ifdown: interface eth0 not configured   Is there something I could do?    EDIT: Content of /etc/network/interfaces    auto lo iface lo inet loopback   When I ifconfig I don't have these lines:    inet addr:192.168.1.5  Bcast:192.168.1.255  Mask:255.255.255.0 inet6 addr: fe80::219:5bff:fe5e:a5e/64 Scope:Link   In the Terminal typed:  sudo gedit /etc/network/interfaces   and then added these lines in the bottom:  auto eth0 iface eth0 inet dhcp   then I typed:    sudo ifdown eth0 sudo ifup eth0   when I send the second command I received the below message:    No DHCPOFFERS received No working leases in persistent database - sleeping   Any help? What is the physical significance of the off-diagonal moment of inertia matrix elements? The tensor of moment of inertia contains six off-diagonal matrix elements, which vanish if we choose a reference frame aligned with the principal axes of the rotating rigid body; the angular momentum vector is then parallel to the angular velocity. But while considering the general case, what are the off-diagonal moment of inertia matrix elements? That is, do they have any physical significance as [say] the components of a vector? Or is it merely a mathematical construction with no definite physical meaning (which seems rather wrong to me)?  A similar thread exists  but they are more interested in the principal axes of the body. It also says:     The physical significance of the off-diagonal components is that   you're using a coordinate system not aligned with the principal   directions of the object. They tell us nothing interesting about the   object itself.   Is that all or is there more to it, perhaps related to properties of tensors in general? Let $G$ be a connected planar graph with a planar embedding where every face boundary is a cycle of even length. Prove that $G$ is bipartite.  Any hints/tips will be greatly appreciated. Combine the first two commits of a Git repository? Suppose you have a history containing the three commits A, B and C:  A-B-C   I would like to combine the two commits A and B to one commit AB:  AB-C   I tried  git rebase -i A   which opens up my editor with the following contents:  pick e97a17b B pick asd314f C   I change this to  squash e97a17b B pick asd314f C   Then Git 1.6.0.4 says:  Cannot 'squash' without a previous commit   Is there a way or is this just impossible? How to dump a man page? It is very easy to put a tilde over the symbol in math mode. Simply type $\tilde{a}$. But how to put a tilde under the symbol in math mode? Can I download the Winter Bash hats for use as profile picture? Can I download  and use them as my profile picture?  I actually like some of them, and want to keep them for future use. Control Cycles/Eevee material nodes and material properties using python? Choosing between LM and GLM for a log-transformed response variable I'm trying to understand the philosophy behind using a Generalized Linear Model (GLM) vs a Linear Model (LM). I've created an example data set below where:  $$\log(y) = x + \varepsilon $$   The example does not have the error $\varepsilon$ as a function of the magnitude of $y$, so I would assume that a linear model of the log-transformed y would be the best. In the example below, this is indeed the case (I think) - since the AIC of the LM on the log-transformed data is lowest. The AIC of the Gamma distribution GLM with a log-link function has a lower sum of squares (SS), but the additional degrees of freedom result in a slightly higher AIC. I was surprised that the Gaussian distribution AIC is so much higher (even though the SS is the lowest of the models).   I am hoping to get some advice on when one should approach GLM models - i.e. is there something I should look for in my LM model fit residuals to tell me that another distribution is more appropriate? Also, how should one proceed in selecting an appropriate distribution family.   Many thanks in advance for your help.  [EDIT]: I have now adjusted the summary statistics so that the SS of the log-transformed linear model is comparable to the GLM models with the log-link function. A graph of the statistics is now shown.  Example  set.seed(1111) n &lt;- 1000 y &lt;- rnorm(n, mean=0, sd=1) y &lt;- exp(y) hist(y, n=20) hist(log(y), n=20)  x &lt;- log(y) - rnorm(n, mean=0, sd=1) hist(x, n=20)  df  &lt;- data.frame(y=y, x=x) df2 &lt;- data.frame(x=seq(from=min(df$x), to=max(df$x),,100))   #models mod.name &lt;- "LM" assign(mod.name, lm(y ~ x, df)) summary(get(mod.name)) plot(y ~ x, df) lines(predict(get(mod.name), newdata=df2) ~ df2$x, col=2)  mod.name &lt;- "LOG.LM" assign(mod.name, lm(log(y) ~ x, df)) summary(get(mod.name)) plot(y ~ x, df) lines(exp(predict(get(mod.name), newdata=df2)) ~ df2$x, col=2)  mod.name &lt;- "LOG.GAUSS.GLM" assign(mod.name, glm(y ~ x, df, family=gaussian(link="log"))) summary(get(mod.name)) plot(y ~ x, df) lines(predict(get(mod.name), newdata=df2, type="response") ~ df2$x, col=2)  mod.name &lt;- "LOG.GAMMA.GLM" assign(mod.name, glm(y ~ x, df, family=Gamma(link="log"))) summary(get(mod.name)) plot(y ~ x, df) lines(predict(get(mod.name), newdata=df2, type="response") ~ df2$x, col=2)  #Results model.names &lt;- list("LM", "LOG.LM", "LOG.GAUSS.GLM", "LOG.GAMMA.GLM")  plot(y ~ x, df, log="y", pch=".", cex=3, col=8) lines(predict(LM, newdata=df2) ~ df2$x, col=1, lwd=2) lines(exp(predict(LOG.LM, newdata=df2)) ~ df2$x, col=2, lwd=2) lines(predict(LOG.GAUSS.GLM, newdata=df2, type="response") ~ df2$x, col=3, lwd=2) lines(predict(LOG.GAMMA.GLM, newdata=df2, type="response") ~ df2$x, col=4, lwd=2) legend("topleft", legend=model.names, col=1:4, lwd=2, bty="n")   res.AIC &lt;- as.matrix(     data.frame(         LM=AIC(LM),         LOG.LM=AIC(LOG.LM),         LOG.GAUSS.GLM=AIC(LOG.GAUSS.GLM),         LOG.GAMMA.GLM=AIC(LOG.GAMMA.GLM)     ) )  res.SS &lt;- as.matrix(     data.frame(         LM=sum((predict(LM)-y)^2),         LOG.LM=sum((exp(predict(LOG.LM))-y)^2),         LOG.GAUSS.GLM=sum((predict(LOG.GAUSS.GLM, type="response")-y)^2),         LOG.GAMMA.GLM=sum((predict(LOG.GAMMA.GLM, type="response")-y)^2)     ) )  res.RMS &lt;- as.matrix(     data.frame(         LM=sqrt(mean((predict(LM)-y)^2)),         LOG.LM=sqrt(mean((exp(predict(LOG.LM))-y)^2)),         LOG.GAUSS.GLM=sqrt(mean((predict(LOG.GAUSS.GLM, type="response")-y)^2)),         LOG.GAMMA.GLM=sqrt(mean((predict(LOG.GAMMA.GLM, type="response")-y)^2))     ) )  png("stats.png", height=7, width=10, units="in", res=300) #x11(height=7, width=10) par(mar=c(10,5,2,1), mfcol=c(1,3), cex=1, ps=12) barplot(res.AIC, main="AIC", las=2) barplot(res.SS, main="SS", las=2) barplot(res.RMS, main="RMS", las=2) dev.off()