Help! A math emergency. An ordinary differential equation I think MIT got wrong

Today’s blog entry is for those math lovers out there, and it assumes some mathematical knowledge that I cannot impart to lawyers or readers who haven’t had multivariable calculus. I was brushing up on a part of math called “ordinary differential equations” and I selected the hardest YouTube lecture I could from the Massachusetts Institute of Technology. The very first part of the course contained a statement I think is mathematically false. The professor, who I don’t know at all, said that the ODE  




contained zero solutions. I’m sorry, but I have reason to believe this is flat out wrong. 

In math, there are three cases. The positive case, the zero case, and the negative case. That means in this equation y can equal the set of positive numbers, the zero case, and the negative case. The same goes for x. X can equal the set of positive numbers, the zero case, and the negative case. 

Here, these are independent variables, meaning they do not depend on one and other and have no relationship. But if you put all the cases together and do all the possible combinations of the negative case, the zero case, and the positive case for both x and y that yields a combination of nine solutions that can be graphed on an actual graph. 

I’m too lazy to do a complete proof, so I assume readers can think geometrically like I can. If I’m wrong, I’ll be the first to admit it, but my main point is that if x and y are independent there are 9 sets of solutions that can be graphed on a number line geometrically and not zero solutions. 

Can anyone show me my mistake because I just don’t see it. I’m no expert in probability but 3 x 3 = 9 at least in the ordinary base most humans use and at least if we aren’t using imaginary numbers. To lawyers who do math, can you show me my error? 

ODEs are used for lots of practical things like population growth and were used by Malthus who is relevant to overpopulation. I wish I could see my mistake, but I can’t. I think being proved wrong is fun, so extra credit to any reader who can show me the error of my ways or present a proof, and I’ll post your solution on my blog, or you are welcome to comment if you think I’m incorrect. I’m always the first to admit when I’m wrong and I love being proved wrong. 


-Cortelyou C. Kenney, 6/16/25 6:26 pm PT


CORRECTION: At 7:05 pm I adjusted the equation because due to tech issues, the image got messed up. This should have the correct image, and I still don't see my mistake. I approach math like debate because I'm a former nationally ranked debater, and I'll debate anyone or give credit to anyone who can prove me wrong. It's actually fun and enjoyable. 


NOTE NOT CORRECTION: On 6/17 I rewatched the video and a reader also brought this to my attention. In the current video the teacher did not say there were no solutions and said that it could be solved in other ways, i.e., my way. Truth is stranger than fiction, but to my best recollection the video did not say when I posted last night on 6/16. I'm not prepared to stake my law license on this, but I watched the video closely, and think I would have noticed. I am not making accusations, merely stating the facts to the best of my abilities. 

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