From 2 1st order ODEs to 1 2nd order ODE?

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Discussion Overview

The discussion revolves around the possibility of transforming a system of two first-order ordinary differential equations (ODEs) into a single second-order ODE. Participants explore the implications and methods of such a transformation, including the challenges and conditions under which it may or may not be feasible.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the motivation behind transforming a system of two first-order ODEs into a higher-order ODE and asks how one would solve the resulting equation.
  • Another participant shares a method for deriving a second-order ODE from a system of equations by differentiating and substituting, illustrating with a specific example.
  • A different participant expresses confusion, noting that their understanding was that second-order ODEs could be solved by reducing them to first-order systems, as explained by their lecturer.
  • One participant mentions that the theory of existence and uniqueness of solutions is often simpler in the context of first-order systems, but acknowledges that both methods (characteristic equation and system reduction) are valid.
  • A participant raises a concern about situations where variables cannot be easily isolated, questioning if there is a general method for eliminating variables through implicit differentiation.
  • Another participant provides an example of non-linear equations, demonstrating the complexity involved in transforming them and highlighting the necessity of considering multiple cases when isolating variables.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility and methods of transforming systems of ODEs into higher-order equations. There is no consensus on a general approach, and the discussion remains unresolved regarding the effectiveness of such transformations in various contexts.

Contextual Notes

Participants note that the ability to transform equations may depend on the specific problem and the nature of the equations involved, particularly in non-linear cases where isolating variables can be problematic.

pauperrimo
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Hi,
it is well known that a second order ODe can be transformed into a system of two ODEs through the transformation u=y', v= y.
Is the other way round possible? I mean, I have a system of 2 ODEs and want to transform it into a sucession on higher order problems that can be solved one after the other. Is this possible?
thanx a lot to anyone who can provide some insight!
 
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Why would you want to do that? How would you solve the higher-order ODE then?
 
Actually, Defennder, that was how I first learned to solve systems of equations!

For example, if we have the system x'= x- y, y'= x+ 2y, we can differentiate the first equation again to get x"= x'- y'= x'+ x+ 2y. That still has a "y" in it but from the first equation y= x- x' so we have x"= x'+ x+ 2(x- x')= 3x- x' or x"+ x'- 3x= 0.That's a second order, linear equation with constant coefficients so it can be easily solved. We can then determine y from y= x- x'. You will note that the "characteristic" equation, r2+ r- 3= 0 is exactly the same as the equation for the eigenvalues for the system of equations.
 
That's odd. I thought the reason why 2nd order homogeneous linear ODEs could be solved was because they could be reduced to a system of first order ODEs. At least that's how my lecturer explained why that technique works for 2nd order ODEs.
 
Certainly the theory, existence and uniqueness of solutions for initial value problems, can be simplest done in terms of systems of first order equations. But the standard way of deriving the solution for a higher order linear, homogeneous ODE with constant coefficients, at least when I was learning it (in years "BC"- "before calculators") was to try y= erx as a solution and derive the characteristic equation from that. Of course it can be done either way. I would think that changing to a system of equations and using the matrix form would be just a little more "sophisticated".
 
HallsofIvy said:
Actually, Defennder, that was how I first learned to solve systems of equations!

For example, if we have the system x'= x- y, y'= x+ 2y, we can differentiate the first equation again to get x"= x'- y'= x'+ x+ 2y. That still has a "y" in it but from the first equation y= x- x' so we have x"= x'+ x+ 2(x- x')= 3x- x' or x"+ x'- 3x= 0.That's a second order, linear equation with constant coefficients so it can be easily solved. We can then determine y from y= x- x'. You will note that the "characteristic" equation, r2+ r- 3= 0 is exactly the same as the equation for the eigenvalues for the system of equations.

Thanks for the input! However, what if y and y' cannot be isolated? Is there any general way to get rid of one of them by implicitly deriving or something? I'm afraid that this substitution technique is very problem-dependent.

The motivation is that I have a numerical method based on spline collocation that works very well with derivatives of any order, but can hardly cope with systems. Therefore, an approach seems to go for a single higher order ODE.
 
Yes, it is very problem dependent. For example, if we have the non-linear equations
x'= x- y2
y'= x+ y2,

then, differentiating the first equation, x"= x'- 2yy' and y'= x+ y so we have
x"= x'- 2y(x+ y)= x'- 2xy+ y2

Clearly y2= x- x', but for the xy term we will have to solve for y: y2= x- x' so [itex]y= \pm\sqrt{x- x'}[/itex] and you will have to do the problem in two parts, using the plus and minus signs. There is no way to avoid that with non-linear equations.
 

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