# ODE homogeneous equations w/constant coefficients

1. Sep 29, 2016

### mmont012

1. The problem statement, all variables and given/known data
Find the general solution
y"+3y'+2y=0
2. Relevant equations
y(t) =c_1e^r_1t + c_2e^r_2t

3. The attempt at a solution
a=1 b=3 c=2
r^2+3r+2=0
(r+2)(r+1)=0
r_1=-2
r_2=-1

General solution: y(t) =c_1e^(-2t)+c_2e^(-t)

I was wondering if the order mattered. The answer in the book is y(t)=c_1e^(-t)+c_2e^(-2t)

I don't think that it does, but I want to make sure before continuing and submitting my hw.
If the order DOES matter, how can I get the correct order?

Thank you.

2. Sep 29, 2016

### Staff: Mentor

No, the order doesn't matter
By the way, what you wrote as a relevant equation confused me for a while.
For the two functions, are they $e^{r_1}t$ and $e^{r_2}t$ or are they $e^{r_1t}$ and $e^{r_2t}$?
I know what you mean, but if you write them inline as you did, at the least use parentheses around the exponent, such as e^(r_1t).

3. Sep 29, 2016

### mmont012

Thank you, and sorry about the confusion. I'm using my phone and its acting up. Next time I'll be sure to add ().

4. Sep 29, 2016

### Ray Vickson

Either order is correct because there are not really any rules about that.

However, sometimes authors adopt certain conventions, such as having the most slowly-decreasing functions first, followed by more rapidly-decreasing functions. In that convention, $e^{-t}$ decreases more slowly than $e^{-2t}$, so would come first. In the opposite case of increasing functions, people sometimes want the most rapidly-increasing functions to come first, in part because they govern the asymptotic large-$t$ behavior. So, if you had increasing functions $e^{t}$ and $e^{2t}$, some people might write the $e^{2t}$ first.

However, as I said, there really are no rules, and not everyone subscribes to the type of conventions I have mentioned.