ODE homogeneous equations w/constant coefficients

[mmont012]

Homework Statement

Find the general solution
y"+3y'+2y=0

Homework Equations

y(t) =c_1e^r_1t + c_2e^r_2t

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.

 

[Mark44]

Homework Statement

Find the general solution
y"+3y'+2y=0

Homework Equations

y(t) =c_1e^r_1t + c_2e^r_2t

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)

No, the order doesn't matter

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?

By the way, what you wrote as a relevant equation confused me for a while.

c_1e^r_1t + c_2e^r_2t

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).

 

[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 ().

 

[Ray Vickson]

Homework Statement

Find the general solution
y"+3y'+2y=0

Homework Equations

y(t) =c_1e^r_1t + c_2e^r_2t

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.

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.