Reducing and increase of order and ODE

[Bruno Tolentino]

Two questions:

First: If is possible to reduce the order of an ODE increasing the number of equations, so, is possible do the inverse patch? In other words, is possible reduce the number of equations of a system of ODE increasing the order?

Second: This technique of reducing and increasing of order is applicable to a PDE/system of PDE?

 

[jedishrfu]

I think certain conditions apply like the PDE must be linear so that separation of variables can be applied some of which is described toward the end of this article:

https://en.m.wikipedia.org/wiki/Partial_differential_equation

 

[HallsofIvy]

Even in with linear equations, whether or not it is separable depends upon the geometry of the situation. An equation may be separable in one coordinate system and not in another.

 

[epenguin]

You can certainly do that in the case where you did the opposite recently https://www.physicsforums.com/threads/system-of-ode-of-second-order.825408/ . More than one way around but if you differentiate one of the equations you will find you have enough to eliminate y and y' and finish with an 2nd order d.e. in x. Then with no added effort with one in y too. In general n linear d.e.s with constant coefficients like that you can get one n'th order one in one variable. In fact you can get n of them. But if you are interested in solutions you only need to get one, and the other solutions are merely a renaming of terms.