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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:

  1. Solve the initial value problems
  2. Determine the interval where the solution exists
  3. 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: $\lim_{t\to 0^+}y(t)$ ?

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up vote 1 down vote accepted

It seems for the second case, both numerator and denominator are continuous at $0$ and evaluate to $2-e <0$ and $0$, respectively, so the quotient should be $-\infty$.

For the first case, the top diverges to $+\infty$ and the bottom is finite, so the result diverges to $\infty$.

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