I had an differential equation exam last semester. There was a question I never found out how I could answer. At a part of the question I faced:
$\frac{u'''}{u''} = \cot(x)u+x$
And it was asked to make it first-order Linear and then solve it. I simply wrote
$\frac{u'''}{u''}= u'$. Then wrote it like $u' - \cot(x)u = x$, but I know the first part is not correct at all. I never had the chance to ask the teacher or anyone what should one do in this situation?
I think I had a confusion and it's root is the notation when you could write $\frac{dy}{dx}$ as $(D)y$ so $\frac{d^3y}{d^3x}$ could be $(D^3)y$.
Is there a way to lower this equations order?
$\frac{d^3 y}{d x^3}$ doesn't represent $\frac{dy}{dx} \frac{dy}{dx} \frac{dy}{dx}$. It represents the iterated operator of differentiation.
$\frac{d^3 y}{d x^3} = \frac{d }{d x} ( \frac{d }{d x} ( \frac{d }{d x} ( y)))$
You can't lower the order of differentiation, that's the whole point of differential equation. You can do some tricks like change of variable $y=u'$ and see how it goes, but it requires solving for $y$, then for $u$ which is strictly equivalent.