Converting a Third-Order Differential Equation into a Vector System?

[johnaphun]

Homework Statement

Convert the differential equation for x,

x''' + 2(x''²) = 0

Into a system of first order differential equations. Put the system in vector form

Homework Equations

The Attempt at a Solution

I'm able to do this for simpler DE's but I can't seem to find an answer for this one. Do i need to do anything different because of the 3rd derivative?

 

[tiny-tim]

hi johnaphun!

what's the difficulty?

put a = x'' and solve, then put v = x' and solve

 

[HallsofIvy]

Same thing except that I tend to prefer to use letters near the beginning of the alphabet, like "a", to represent constants, letters near the end, like "x", to represent variables.

Since your equation is $x'''+ 2(x'')^2= 0$, let y= x' and z= y'= x'' so that x'''= z'. Now, $x'''+ 2(x'')^2= z'+ 2z^2= 0$, $y'= z$, and $x'= y$.

 

[tiny-tim]

Same thing except that I tend to prefer to use letters near the beginning of the alphabet, like "a", to represent constants, letters near the end, like "x", to represent variables.

Normally, I'd agree! … but this equation looked to me like a dynamical equation, with x being distance, so I preferred the familiar form of x' = v, v' = a.