Comment-answer, but too long for a comment:
I think you are thinking about this wrong.
Tests are some of the MOST valuable hours in a course. They are high stakes performances (like in music or sports). Preparation for them drives a lot of learning. Then the actual execution and subsequent feedback is often much more valuable training than routine drill problems, because of caring about the result. You should maximize test utility versus another hour of lecture (which probably has less learning impact even than an hour of them working drill problems).
In addition, by using a take home you are trying to get an extra hour of their time. In addition, take homes are very hard to control from assistance (book, roommate, online). Can you really expect them to be rigid about a time limit when taking a test at home? If unlimited, then now you're taking more of their time.
In addition, take home tests often lead to instructors giving non ideal questions (ones that are too much like research projects for instance). This is not the case here--your question is fine. However, your urge to ask some sort of strange, hard to cheat on question may already be driving you away from what would be the normal best question in a controlled setting.
In general, ODE books have too much content for a one quarter or one semester course. It is very normal not to cover it all. So if you're already shaving some lessons off, just shave one more off. And learn the core of the topic well. And good tests are key to good learning. Don't shave this 2nd order ODE with constant coefficients off. It's the key part of the course! Very high gain. But surely there is some other lesson (existence, Wronskians, predator prey, transforms, Sturm Louisville, series, etc.) you can skip to allow use of class time for examination.
And I love all the topics and would love to have you cover everything. But you need to prioritize. You got a lifeboat with 15 spots and 25 people, pick the best 15 to live. 2nd order constant coefficient ODE, with a forcing function, will be seen all over the place in physics, engineering, chemistry applications. So it lives. Some other topics go in the water. And the test time needs to live. It's "high gain".
In theory, you could cover more content by telling the kids to learn a lesson sans lecture. But I get the impression you're not training superstars here. They will be unhappy without the scaffolding of some lecture time and Q&A time on the topic. So, just treat this as an optimization problem and do a little less, but do it well.
[Moderator may cut my other answer.]