Systems of Differential Equations and higher order Differential Equations. I've seen how one can transform a higher order ordinary differential equation into a system of first-order differential equations, but I haven't been able to find the converse. Is it true that one can transform any system into a higher-order differential equation? If so, is there a general method to do so?

Dulce Cantrell 2022-09-11 Answered
Systems of Differential Equations and higher order Differential Equations.
I've seen how one can transform a higher order ordinary differential equation into a system of first-order differential equations, but I haven't been able to find the converse. Is it true that one can transform any system into a higher-order differential equation? If so, is there a general method to do so?
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Answers (2)

cerfweddrq
Answered 2022-09-12 Author has 15 answers
Step 1
If I am understanding your question, you just would reverse the process on the last equation from the system.
An n t h order differential equation can be converted into an n-dimensional system of first order equations.
There are various reasons for doing this, one being that a first order system is much easier to solve numerically (using computer software) and most differential equations you encounter in “real life” (physics, engineering etc) don’t have nice exact solutions.
Step 2
If the equation is of order n and the unknown function is y, then set:
x 1 = y , x 2 = y , , x n = y n 1 .
Note (and then note again) that we only go up to the ( n 1 ) s t derivative in this process. Lets do an example in both directions

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peckishnz
Answered 2022-09-13 Author has 2 answers
Step 1
Forward Approach
(1) y ( 4 ) 3 y y + sin ( t y ) 7 t y 2 = e t
Let: x 1 = y , x 2 = y , x 3 = y , x 4 = y and substitute into (1), yielding:
- x 1 = y = x 2
- x 2 = y = x 3
- x 3 = y = x 4
- x 4 = y ( 4 ) = 3 y y sin ( t y ) + 7 t y 2 + e t = 3 x 2 x 4 sin ( t x 3 ) + t x 1 2 + e t
Step 2
Looking at the last equation from the system, we let: y = x 1 , y = x 2 , y = x 3 , y = x 4 and substitute into the system's last equation above, yielding:
- y ( 4 ) = 3 y y sin ( t y ) + 7 t y 2 + e t

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