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which satisfy the simultaneous equations
and
where 
and 
are integers such that 
and 
.
, s.t. ecuation: 
have 2 real solutions .
, 
. Are they isolated?
be a ring with unity such that for every 
there exist 
such that 
. Prove that
then 
such that 
then the result from a) may no longer hold.
and 
be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence 
is infinite.
![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
, where 
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
![\[
I = 2 \int_{0}^{\pi/4} \frac{1}{\cos 2\phi} \cdot 2 \ln(\cot \phi) \cdot 2 \, d\phi
\]](http://latex.artofproblemsolving.com/7/a/8/7a82b62346bf5636f262c4b27f3a4cf462d8deba.png)
![$$\int_0^{\pi/2} \Bigl[
\log\bigl(\sqrt{5}-\sin\theta+1\bigr)
+\log\bigl(\sqrt{5}+\sin\theta-1\bigr)
-\log\bigl(\sqrt{5}-\sin\theta-1\bigr)
-\log\bigl(\sqrt{5}+\sin\theta+1\bigr)
\Bigr]\,d\theta.
$$](http://latex.artofproblemsolving.com/a/f/f/affd7cb3d94d4fbcab6f9912e71ebcffeb86d960.png)
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
