Context: My friend (The same one who gave me this ODE) originally challenged me to obtain the general solution to this ODE: $$\frac{d^3 y}{dx^3}+\alpha x\frac{dy}{dx}+\beta y=0 \tag{1}$$ Where $\alpha,\beta \in \mathbb{R}$. I could not figure out any substitution without the use of power seri...

In the book "The theory of Ordinary Differential Equations" I'm having trouble understanding $Lemma(1.6)$ which assets Lipchitz Continuity defined below, specifically the gap in my understanding is within the operation taken within $(2.)$,specifically speaking the specifics of my question can be ...

In the book "The theory of Ordinary Differential Equations" I'm having trouble understanding $Lemma(1.6)$ which assets Lipchitz Continuity defined below, specifically the gap in my understanding is within the operation taken within $(2.)$,specifically speaking the specifics of my question can be ...

Define $s=-\frac{\alpha x^n}{n^{n-1}}$ and introduce Euler operators $\vartheta=s\frac{d}{ds}$ and $\delta=x\frac{d}{dx}=n\vartheta$. Lemma. $x^n\frac{d^n}{dx^n}=\delta(\delta-1)(\delta-2)\ldots (\delta-n+1)$. Corollary 1. $\frac{d^n}{dx^n}=n^n\vartheta\left(\vartheta-\frac1n\right)\left(\varth...