Show that the origin is a critical point (linear differential equation)

[JJBladester]

Homework Statement

Show that the origin (0,0) is a critical point. Write the linear differential equation in operator format and solve.

Homework Equations

x'' + 10x' + 25x = 0

The Attempt at a Solution

I am not sure how to show that the origin is a critical point (without using a graph).

As for solving the 2nd-order linear differential equation, here's what I did:

x'' + 10x' + 25x = 0

Auxillary equation: m² + 10m + 25 = 0
Roots: m₁ = m₂ = -5

General solution: x = c₁e^-5x + c₂xe^-5x

I believe my general solution is correct, but am not sure if I solved it by "operator method".

 

[tiny-tim]

Hi JJBladester!

Show that the origin (0,0) is a critical point. Write the linear differential equation in operator format and solve.

x'' + 10x' + 25x = 0
…
General solution: x = c₁e^-5x + c₂xe^-5x

I believe my general solution is correct, but am not sure if I solved it by "operator method".

Yes, that's the correct general solution …

I think by operator format they just mean D² + 10D + 25 = 0

But I don't quite understand what is meant by (0,0) …

does that mean that there is an initial condition of x' = 0 at x = 0?

if so, (0,0) is a critical point just by looking at x'' + 10x' + 25x = 0 … you don't need to solve it.

 

[JJBladester]

I don't quite understand what is meant by (0,0) …

does that mean that there is an initial condition of x' = 0 at x = 0?

if so, (0,0) is a critical point just by looking at x'' + 10x' + 25x = 0 … you don't need to solve it.

My professor said I should:

"Change the 2nd order DEQ into a system of first order DEQ's.
Basically...Let y=dx/dt and then substitute... When you get these two equations, set both derivatives equal to zero and solve for the x and y value that solves the system."

I'm still a little confused what that means.

Here is my attempt:

Let y = x', y'=x''

So x'' + 10x' + 25x = 0 becomes y' + 10y + 25x = 0.

-10y - 25x = 0
y = 0
-25x = 0
x = 0

 

[tiny-tim]

hmm … I've no idea what he means …

unless maybe he means put y = x' + 5x …

then y' + 5y = 0 …

dunno … as I said in my last post, I don't really understand the question

 

[HallsofIvy]

Let y= x'. Then x'' + 10x' + 25x = 0 becomes y'+ 10y+ 25x= 0 and we have the system of equations x'= y, y'= -10x-25y. A "critical point" is (x,y) such that x'= 0 and y'= 0.

 

[tiny-tim]

Let y= x'. Then x'' + 10x' + 25x = 0 becomes y'+ 10y+ 25x= 0 and we have the system of equations x'= y, y'= -10x-25y. A "critical point" is (x,y) such that x'= 0 and y'= 0.

why can't we just put x'' = x' = 0 in the original equation, then, and forget about y?