I would consider to add two items to the list, both from a systems slant:
- Predator prey relations. The behavior can be graphically investigated, but the actual solution function is not analytically soluble. Any solid ODE book will cover this. E.g. Speigel's
https://www.amazon.com/Applied-Differential-Equations-Murray-Spiegel/dp/0130400971#customerReviews
- Xenon transients in a nuclear reactor. Long term equilibria can be calculated (limited of time as infinity), but the function of time is not analytically soluble. The standard problem is response to a power change (assume a homogenous reactor). In actuality, very large civilian reactors can have strange oscillations in the geometry and time dimension. But I would avoid this as it is a PDE and too complicated.
See: http://mafija.fmf.uni-lj.si/seminar/files/2016_2017/Seminar_-_Xenon_oscillations_rev3.pdf
[Edit adding derivation]
Xe is a reactor "poison" (absorbs neutrons in competition with uranium). Iodine is a precursor of Xe (look at the periodic table.) For each of them, the $\frac{\mathrm{d}}{\mathrm{d}t}$Concentration = production rate - loss rate. Xe has two PR and two LR. I (iodine) has one of each.
$$\frac{\mathrm{d}I}{\mathrm{d}t}=PR-LR=aPWR-bI$$
Addition is correlated to power as a certain percent of the fissions generate this fission product. Loss rate is just exponential decline.
$$\frac{\mathrm{d}}{\mathrm{d}t}Xe=PR-LR=PR_1+PR_2-LR_1-LR_2$$
$$\frac{\mathrm{d}}{\mathrm{d}t}Xe=cPWR+bI-dXe-ePWR$$
Xe has an extra production term as it is produced directly from fission and also from Iodine decay. It has an extra loss term as it goes away both by decay and by "burnout" (the poisoning reaction).
Note, where I write PWR, it really should be the local neutron flux. Which is related to overall power but not perfectly linearly (there is leakage at the reactor border for instance).
Note, that the Xe situation while it has the trappings of mechE (power reactor) and physics (radioactive decay), it is really a chem E problem in sheep's clothing. As you are looking at concentration. Many feed/bleed problems with salt tanks and the like live in ODE world. And I would consider to included one or two. Have a lot of application in chemistry, environment, etc. I also find concentration somewhat intuitive (more so than bending beams), but maybe that's me.
Comments:
I'm not an expert on this topic, but FWIW, your (2) is the very standard core of analytical ODE course. Unless the forcing function is quite unusual, the problem is soluble.
I believe your (3) is typically a PDE, not ODE, problem. Perhaps some of the others as well. However there area lot of good PLATE (not rod) heat transfer problems that are ODEs. (I seem to recall them as analytically soluble, but there may be some that are not.)