I'm given the ODE:


I try to solve it regularly and I get $y(x) = \sqrt{4+x}(\ln|4+x|+C)$ for some constant $C$. I'm also not entirely sure I got the math correctly, however, the solution has two answers:

$y(x) = \bigg(\sin^{-1}(\frac{x}{4}) +a\bigg)\sqrt{4+x}$

if $x > -4$, and

$y(x) = \bigg(\ln|x+\sqrt{x^2 -16}| + a\bigg) \sqrt{|x+4|}$

if $x \leq -4$.

Looking at these 2 solutions, it looks like my original answer is incorrect, but I'm unsure why there are two different answers for 2 intervals. I'm guessing because there is a discontinuity in the denominator in the right-hand side, but I don't know how to deal with it. Can someone explain this?



without RHS


which gives




now, let us look for a particular solution of the form


if we replace, we get

$$(4+x)\frac{\lambda'(x)}{\sqrt{|x+4|}}= \frac{x+4}{\sqrt{4-x}}$$



which could be integrated at$(-\infty,-4]$ and at $[-4,4)$.


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