Can I separate a differential equation?

[sliken]

Given the following differential equation

x*x''+(x')^2+y*y''+(y')^2=C

where C is a constant and all differentiation is with respect to time


Can i equal the first and second parts of the equation into different constants and solve separately?, meaning solving the system

x*x''+(x')^2=k^2
y*y''+(y')^2=C-k^2

 

[Fightfish]

No. Why would you expect that to hold true?
The equation can be slightly simplified though.
Think of how $x x'' + (x')^2$ can be rewritten.

 

[Dustinsfl]

You can write it has first order equations but I don't know if that is your aim.

 

[vela]

Given the following differential equation

x*x''+(x')^2+y*y''+(y')^2=C

where C is a constant and all differentiation is with respect to time


Can I equal the first and second parts of the equation into different constants and solve separately?, meaning solving the system

x*x''+(x')^2=k^2
y*y''+(y')^2=C-k^2

[STRIKE]Yes. Why do you suppose you can do that?[/STRIKE] (Never mind.)

 

[LCKurtz]

Can i equal the first and second parts of the equation into different constants and solve separately?, meaning solving the system

x*x''+(x')^2=k^2
y*y''+(y')^2=C-k^2

Yes. Why do you suppose you can do that?

I don't see why the constant must be positive for the $x$ equation. But, more to the point, I am unconvinced that the answer is yes. It isn't like the eigenvalue situation you get in separation of variables in partial DE's because the two sides have different independent variables. Or maybe that isn't what your reasoning is.

 

[Mark44]

No. Why would you expect that to hold true?
The equation can be slightly simplified though.
Think of how $x x'' + (x')^2$ can be rewritten.

I like this idea.

 

[vela]

I don't see why the constant must be positive for the $x$ equation. But, more to the point, I am unconvinced that the answer is yes. It isn't like the eigenvalue situation you get in separation of variables in partial DE's because the two sides have different independent variables. Or maybe that isn't what your reasoning is.

Yeah, you're right. Never mind my earlier post.