let the integral equation is  $$\phi (x)-\frac{e}{2}\int_{-1}^{1}xe^{t}\phi(t)dt=f(x)$$  then  1).there exist a continous function $f:[-1,1]\to (0,\infty)$ for which solution exist. 2).there exist a continous function $f:[-1,1]\to (-\infty,0)$ for which solution exist. 3).for $f(x)=e^{-x}(1-3x^2)$ ,a solution exist 4).for $f(x)=e^{-x}(x+x^3+x^5)$ ,a solution exist  how to solve this question ,i can only solve simple questions in integral equations but this very tough , $\text{EDIT}$ After so much of reading of integral equation,i try to solve the question $\textbf{the solution i tried}$-$$\phi (x)=\frac{e}{2}\int_{-1}^{1}xe^{t}\phi(t)dt+f(x)$$ $$\phi (x)=\frac{e}{2}x\int_{-1}^{1}e^{t}\phi(t)dt +f(x)$$ $$\phi (x)=\frac{e}{2}xC+f(x)    \;\;\;\;\;\;\;....1$$ where $C=\int_{-1}^{1}e^{t}\phi(t)dt$ $\textbf{from (1)}$ $$C=\int_{-1}^{1}e^{t}(\frac{e}{2}tC+f(t))$$ $$C=\int_{-1}^{1}e^{t}\frac{e}{2}tCdt+\int_{-1}^1 e^tf(t)dt$$ $$C=\frac{e}{2}C\int_{-1}^{1}e^{t}tdt+\int_{-1}^1 e^tf(t)dt$$ further doing calculation i get $$C=C+\int_{-1}^1 e^tf(t)dt$$ $$\implies \int_{-1}^1 e^tf(t)dt=0$$ after that i pick the option  'c' and 'd' and they satisfied the integral so they are sutiable for $f(x)$ and my option are also right but still don't know how to work for option 'a' and 'b' please help. Thankyou
Consider the integral equation $\phi(x)-\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t=f(x) .$ Then  there exists a continuous function $f:[-1,1] \rightarrow(0, \infty)$ for which solution exists there exists a continuous function $ f:[-1,1] \rightarrow(-\infty, 0)$ for which solution exists for $f(x)=e^{-x}\left(1-3 x^{2}\right)$, a solution exists for $f(x)=e^{-x}\left(x+x^{3}+x^{5}\right),$ a solution exists  what I tried $f(x)=f(x)+\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t$ $\phi(x)=f(x)+\frac{e x}{2} \int_{-1}^{1} e^{t} \phi(t) d t$ $ \begin{array}{l}L e t \quad c=\int_{-1}^{1} e^{t} \phi(t) d t \\ \Rightarrow \phi(x)=f(x)+\frac{x e}{2} c\end{array} $ $c=\int_{-1}^{1} e^{t}\left(f(t)+\frac{c e}{2} t\right) d t$ $c=\int_{-1}^{1} e^{t} f(t)+\frac{\operatorname{ce}}{2} \int_{-1}^{1} t e^{t} d t$ $c=\int_{-1}^{1} e^{t}f( t)+\frac{\operatorname{ce}}{2}\left[t e^{t}-e^{t}\right]_{-1}^1$ $c=\int_{-1}^{1} e^{t} f(t)+\frac{c e}{2}\left(2 e^{-1}\right)$ $c=\int_{-1}^{1} e^{t} f(t) d t+c$ $\int_{-1}^{1} e^{t} f(t) d t=0$ Option (3) and (4) satisfies this Relation so, (3) and (4) seems true  to me.is i am right? I am stuck  for option (1) and (2)  is there any result or theorem ?
Why couldn't River Song's memories bring back The Doctor in the rebooted universe? In the season 5 finale of Doctor Who, in the wedding ceremony of Amy, River Song showed up outside the door. Moments later, we discovered that she gifted Amy her blank diary which was supposed to contain information about The Doctor. It means River Song remembered The Doctor.  Memories of The Doctor was able to bring The Doctor back to existence in the rebooted Spacetime. Why couldn't River Song's memories work? Avoid trailing zeroes in printf() "who" vs. "whom" and when to omit the relative pronoun This is the boy who was asked by the teacher to stay back after school. This is the boy whom the teacher asked to stay back after school. This is the boy the teacher asked to stay back after school.   Are the sentences above correct ? How to properly write multiple authors in bibtex file? Applications Don't Show Up in Spotlight Spotlight has stopped showing any results for my Applications. I've tried reindexing and removing the index so it rebuilds it. No change. I've tried adding Applications to the Privacy tab and removing it, no change.  I tried repairing disk permissions and redoing the above, no change. I've tried removing everything from the index except Applications and then I just get nothing for any search at all (except dictionary entries).  I tried adding a symlink in my homedir to Applications and reindexing, but no change.  Any ideas on what to do?  I'm running Snow Leopard.  This is driving me crazy!  Update: I've noticed that when I start a reindex with sudo mdutil -E / and then immediately do a Spotlight search for an app that the app shows up temporarily until Spotlight gets disabled due to active indexing.  After the indexing is done, the app entries go away. Upgrade from MaaS 2.6.2 to MaaS 2.7 Diamond is not shown for mod name under wiki posts If a diamond mod is an author of a wiki post then the diamond is not displayed besides the mod name. E.g. :        Even when the name refers to the user profile page (in case of more then one author) like in :        Is it intentional or a bug? Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,  and $\mu_X \approx 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $\mu_X \approx 0$ of a more . How to correctly install Minecraft, including setting any needed packages or apps to install it and to troubleshoot the most common problems that Ubuntu users face when trying to use Minecraft.  I am trying to install and run Minecraft without first having to install the Oracle version of Java. I want to use the OpenJDK version that comes with Ubuntu and as a last resort, use the Oracle one.  I also want to know a list of problems related to Minecraft in Ubuntu and the possible solutions to them.  A step by step guide on how to install Minecraft would also help. How do I combine/exclude multiple tags from a search? I want to search for several related tags at once. When I type multiple tag-names in the search box, only one gets selected and I have to sift through a horde of non-pertinent questions.   E.g. 1: I want to find all questions where 'java' AND 'netbeans' AND 'tomcat' all are defined per question.  E.g. 2: (this may be a bit much to hope for) I want to find all questions where 'java' AND 'netbeans' BUT_NOT 'tomcat' is defined.  Alternatively, is there a way to search through the previous search results?  Is this possible? I've been reading a lot about closures and I think I understand them, but without clouding the picture for myself and others, I am hoping someone can explain closures as succinctly and clearly as possible.  I'm looking for a simple explanation that might help me understand where and why I would want to use them. Prefer composition over inheritance? Why prefer composition over inheritance?  What trade-offs are there for each approach?  When should you choose inheritance over composition? Can anyone explain precisely what IOWait is? Understanding central limit theorem What is wrong with the following sentence:      The Central Limit Theorem  implies that, as the sample size grows, the error distribution approaches normality.   Am I correct by saying that it should in stead state the MEAN of the sample error approaches zero as sample size grows? Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds? How to list all installed packages I'd like to output a list of all installed packages into a text file so that I can review it and bulk-install on another system.  How would I do this? Could you please help me computing    $\Big\lfloor\underbrace{\sqrt{2017+\sqrt{2017+...+\sqrt{2017}}}}_\text{2017 roots}\Big\rfloor$? Can I be admitted to graduate school in a different field from my degree?  Specific cases include:   If I've taken plenty of advanced courses in field X in the process of completing a degree in another field, can I apply to graduate school in X? What if I haven't taken many courses in X, but I have acquired a good grasp of X through self-study or working in a related field? What if I've never studied X, but I have done very well in an unrelated field?  Could I be admitted to graduate school in X on the basis of general intellectual promise, and then make up the missing background after enrollment?   Note that this question is an attempt to provide a comprehensive answer, to avoid the need for a profusion of field-specific questions on this topic (see the associated ). Please feel free to edit the question or answer to improve them. Please refer the following questions asked elsewhere on this site:       I am a native Hindi speaker; Hindi has a plethora of terms referring to relationships. To take a trivial example, the term uncle may refer to father's elder brother, father's younger brother, mother's brothers, father's sister's husband, mother's sister's husband, all of which have specific addresses in Hindi.  I discussed the same with one of my teachers. She had the opinion that the development of language mirrors the cultural moorings of the society that uses the language. Now, since all of the above relationships have distinct status and reverence in the Hindi speaking society, that calls for different addresses. That may well not be the case in English speaking societies.  Is that correct? What other linguistic reasons may account for such a dearth of vocabulary in English in this area? I have problems transferring binary programs to a micro-controller prototype board when using ubuntu/kubuntu to mount the board via usb. With MacOS and Windows there are no problems. I have found the writing from Ubuntu definitely changes the written file. I suspect that the write caching is the problem. How can I disable it, or what else might be the problem.