# Thread: Determine r for Explicit Formula

1. ## Determine r for Explicit Formula

Determine what the value(s) of r has to be in order to find explicit formulas (that is, formulas without integrals) for solutions to $\displaystyle \displaystyle \frac{dy}{dt}=t^ry+4$. Determine what the general solutions are for each value of $\displaystyle r$.

2. Just by looking at it, $\displaystyle -1\leq{r}\leq{0}$ have a few good ones.

Do you see why, say 2, will not work?.

If r=2, then you have an integrating factor of $\displaystyle t^{2}$.

Then $\displaystyle e^{\int{t^{2}}dt}=e^{\frac{t^{3}}{3}}$

Which is not integrable by elementary means.

Actually, $\displaystyle e^{\int{t^{r}}}dt=e^{\frac{t^{r+1}}{r+1}}$ are generally not easily integrated.
Trying to integrate $\displaystyle \int{e^{\frac{t^{r+1}}{r+1}}}dt$ can be a booger.

Then you have for r=0 $\displaystyle e^{\int{dt}}=e^{t}$

If r=-1, then $\displaystyle e^{\int\frac{1}{t}dt}=t$

And so on. Just something to think about. See if you can find others in the interval that will or will not work.