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?
 A: If I am understanding your question, you just would reverse the process on the last equation from the system.
An $n^{th}$ 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.
If the equation is of order $n$ and the unknown function is $y$, then set:
$$x_1 = y, x_2 = y', \ldots , x_n = y^{n−1}.$$
Note (and then note again) that we only go up to the $(n − 1)^{st}$ derivative in this process. Lets do an example in both directions (practice some where a known system has been converted to such a system and make sure you can work backward).
Forward Approach
$$\tag 1 y^{(4)} - 3y' y'' + \sin(t y'') -7ty^2 = e^t$$
Let: $x_1 = y, x_2 = y', x_3 = y'', x_4 = y'''$ and substitute into $(1)$, yielding:


*

*$x_1' = y' = x_2$

*$x_2' = y'' = x_3$

*$x_3' = y''' = x_4$

*$x_4' = y^{(4)} = 3yy''-\sin(ty'')+7ty^2+e^t = 3x_2x_4-\sin(tx_3)+tx_1^2+e^t$


Backward Approach
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:


*

*$y^{(4)} = 3y'y''-\sin(ty'')+7ty^2+e^t$

