Hi,

I am going through some solved problems, but I don't understand the answers very well... I would really appreciate it if someone could explain me the answers briefly. I know most of the problems ahead, but I don't have much of a clue about the first problem of this chapter.

Thanks a lot for your help!

The problem says:

"In each of Problems 1 through 6 determine intervals in which solutions are sure to exist.

1) $\displaystyle y^{iv}+4y'''+3y=t$

2) $\displaystyle ty'''+(sin(t))y''+3y=cos(t)$

3) $\displaystyle t(t-1)y^{iv}+e^ty''+4t^2y=0$

4) $\displaystyle y'''+ty''+t^2y'+t^3y=ln(t)$

5) $\displaystyle (x-1)y^{iv}+(x+1)y''+(tan(x))y=0$

6) $\displaystyle (x^2-4)y^{vi}+x^2y'''+9y=0$"

ANSWERS:

1) $\displaystyle -\infty<t<\infty$

2) $\displaystyle t>0$ or t<0

3) $\displaystyle t>1, or 0<t<1$, or $\displaystyle t<0$

4) $\displaystyle t>0$

5) $\displaystyle ..., {-3\pi}/2<x<{-\pi}/2, {-\pi}/2<x<1, 1<x<{\pi}/2, {\pi}/2<x<{3\pi}/2, ...$

6) $\displaystyle -\infty<x<-2, -2<x<2, 2<x<\infty$

So well, I understand the first one, and as for the others, I am confused about this:

2) Why can't $\displaystyle t$ be equal to zero?

3) Again, why can't $\displaystyle t$ be equal to 0 or 1?

4) It says that $\displaystyle t$ has to be positive, but why is that the case if $\displaystyle ln(t)$ can take negative numbers? Does that have anything to do with $\displaystyle e^t$?

5) I know tan(x) is not defined when x is equal to $\displaystyle {-3\pi}/2, {-\pi}/2, {\pi}/2, {3\pi}/2$, etc. However, why aren't there any answers when $\displaystyle x=1$?

6) Is this result taken from $\displaystyle (x^2-4)$? If so, why?