I'm studying from the "Elementary Differential Equations and Boundary Value Problems" textbook by William Boyce, Richard Diprima and I have come across a group of problems that asks me to:
- Solve the initial value problems
- Determine the interval where the solution exists
- Determine the behavior of the function while the the variable t approaches the endpoints of the interval
So, the two differential equations that I tried to solve so far are:
$ty'+2y=t^2-t+1,\ \ \ y(1)=\frac12$ and $ty'+y=e^t,\ \ \ y(1)=1$
The solutions that I found seem to check out (there are results at the back of the book):
$y=\frac{3t^2-4t+6+\frac{1}{t^2}}{12},\ \ \ t>0$ and $y=\frac{e^t+1-e}{t},\ \ \ t>0$
My problem lies in the 3rd question of the problem. I take the limit to $\infty$ and it's easy to figure out that as $t\to\infty,\ \ \ y\to\infty$ too (in both cases). I'm taking the $\lim$ of $y$ to do that.
However, I do not know how to calculate the $\lim$ of the functions as $t\to0^+$. If I try to use L'Hopital it obviously won't work since the functions are not even continuous in $t=0$.
Does anyone know how I can figure out what is the: $\lim_{t\to 0^+}y(t)$ ?