# ODE: System of Linear Equations usuing Diff. Operator

1. Nov 22, 2016

### RJLiberator

1. The problem statement, all variables and given/known data
This is an ordinary differential equation using the differential operator.

Given the system:
d^2x/dt - x + d^2y/dt^2 + y = 0
and
dx/dt + 2x + dy/dt + 2y = 0

find x and y equation

y = -3ce^(-2t)

2. Relevant equations

3. The attempt at a solution

We change it into a differential operator equation...

[D^2-1]x+[D^2+1]y = 0
[D+2]x +[D+2]y = 0

We simply cancel values out until we are left with

3x+5y = 0
[D+2]x +[D+2]y = 0

Here we have x = -5y/3 and y = -3x/5

We substitute into the second equation and we can easily find that

x = ce^(-2t)
y = ce^(-2t)

My question is: How does the answer book know that there is a 5 for the x equation and a -3 for the y equation. It seems to me that the constant "c" handles this for me.
Am I doing something wrong to get a less precise answer? Or did I work out this problem correctly.

Thank you.

2. Nov 22, 2016

### eys_physics

What do you mean by "cancel values". This is a bit sloppy notation. Remember that $D$ is an operator.
Your first equation is telling you that $3x=-5y$, which is explaining the factors 5 and -3.

No, this is wrong, since it violates the first equation.

I also recommend that you use Latex to write your equations. It makes them much easier to read.

Last edited by a moderator: Nov 22, 2016
3. Nov 22, 2016

### Staff: Mentor

Yeah, I'm not following what you did, either.How did you go from the 2nd-order DE to 3x + 5y = 0?
I hope it's obvious that these two equations are equivalent.

4. Nov 22, 2016

### lurflurf

$$(\mathrm{D}^2-1)x+(\mathrm{D}^2+1)y=0\\ (\mathrm{D}^2-4)(x+y)+3x+5y=0\\ (\mathrm{D}-2)(\mathrm{D}+2)(x+y)+3x+5y=0\\ (\mathrm{D}-2)0+3x+5y=0\\ 3x+5y=0$$
since the second equation is
$$(\mathrm{D}+2)(x+y)=0$$

you should have two constants
x = c1e^(-2t)
y = c2e^(-2t)

related by
3x+5y=0

5. Nov 22, 2016

### RJLiberator

The key distinction for me in this thread was the relationship of 3x+5y=0. I now see how my answer can be corrected to the final answer as presented in the book.

Thank you guys for the help.

I was very close :p.