 Dulce Cantrell

2022-09-11

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? cerfweddrq

Step 1
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.
Step 2
If the equation is of order n and the unknown function is y, then set:
${x}_{1}=y,{x}_{2}={y}^{\prime },\dots ,{x}_{n}={y}^{n-1}.$
Note (and then note again) that we only go up to the $\left(n-1{\right)}^{st}$ derivative in this process. Lets do an example in both directions peckishnz

Step 1
Forward Approach
$\begin{array}{}\text{(1)}& {y}^{\left(4\right)}-3{y}^{\prime }{y}^{″}+\mathrm{sin}\left(t{y}^{″}\right)-7t{y}^{2}={e}^{t}\end{array}$
Let: ${x}_{1}=y,{x}_{2}={y}^{\prime },{x}_{3}={y}^{″},{x}_{4}={y}^{‴}$ and substitute into (1), yielding:
- ${x}_{1}^{\prime }={y}^{\prime }={x}_{2}$
- ${x}_{2}^{\prime }={y}^{″}={x}_{3}$
- ${x}_{3}^{\prime }={y}^{‴}={x}_{4}$
- ${x}_{4}^{\prime }={y}^{\left(4\right)}=3y{y}^{″}-\mathrm{sin}\left(t{y}^{″}\right)+7t{y}^{2}+{e}^{t}=3{x}_{2}{x}_{4}-\mathrm{sin}\left(t{x}_{3}\right)+t{x}_{1}^{2}+{e}^{t}$
Step 2
Looking at the last equation from the system, we let: $y={x}_{1},{y}^{\prime }={x}_{2},{y}^{″}={x}_{3},{y}^{‴}={x}_{4}$ and substitute into the system's last equation above, yielding:
- ${y}^{\left(4\right)}=3{y}^{\prime }{y}^{″}-\mathrm{sin}\left(t{y}^{″}\right)+7t{y}^{2}+{e}^{t}$

Do you have a similar question?