I'm working on the problems in the book "Asymptotic Methods of Differential Equations", by Roscoe White. It's a pretty legit book, and all the problems are quite non-trivial and very rich. However, two of the problems in Chapter 2 (which talks about exact solutions and has quite a bit on delta functions and Greens functions), are really vexing me at the moment. They are stated quite unceremoniously as:
$6.$ solve $x^3 y' = 2y^2 + 3x^2 = 0$
$7.$ solve $y' = 2xy + y = 0$ for $y = y(x)$.
(I triple checked the two lines above, they are typed exactly as written in the pages of the book)
Now since the chapter that this problem set was placed in talks extensively about Greens functions, it seems that I'm supposed to construct some crazy solution out of delta functions and step functions. However, after trying for about half an hour, I tried checking these equations for consistency, and noted that if you differentiate the second equation in problem $6$ and multiply by $x^3$, it seems that there is a contradiction. Similarly, if you differentiate the second equation in Problem 7, it seems that the only possible solution is the trivial $y = 0$.
Now if this were some random problem set cobbled together by a TA, then I would probably just move on and assume it was sloppiness on the part of whoever created these. However, the context of the problems tells me that I'm missing something, and there may an non-trivial solution to these. Can anyone take a look and tell me what I'm supposed to do here? Thanks.