From Tenenbaum's Ordinary Differential Equations", ch. 1, Exercise 2e:
Prove that the function is a solution of the differential equation, and state the common interval for which solution and differential equation make sense.
Differential Equation: $xy^{\prime} = 2y$
Function: $y=x^{2}$
Proving that $y=x^{2}$ satisfies the differential is fine, I got that part. The book says the set on which $y=x^{2}$ is a solution is: $x \neq 0$.
I see that if you rearrange the differential to $y^{\prime}=\frac{2y}{x}$, why x cannot equal zero. But how does one know to rearrange the differential relation like this in the first place?
IOW, why is the division by x introduced into the original relation in order to get $x\neq 0$?
And what if this were a higher order differential, of order n for example: $F(x,y,y^{\prime},\cdots,y^{(n)})=0$. Is the rule/convention that one should express $y^{(n)}$ (the highest order differential) as an explicit function, with $x,y,y^{\prime},\cdots,y^{(n-1)}$ as independent variables?
Best Answer
The general solution is actually
$$\int \frac{1}{y}dy=2\int\frac{1}{x}dx\\ \ln y = 2\ln x +C\\ y=e^Ce^{2\ln x}=Cx^2$$
Assuming $x_0\neq0$, you can choose any initial point $(x_0,y_0)$ and find a unique solution of this form by letting $$C=\frac{y_0}{x_0^2}$$ For example, the solution of the differential equation that passes through $(4,2)$ will be $y=\frac{x^2}{8}$.
But if $x_0=0$, then we have $$xy^\prime=2y\\ 0=2y\\ y=0$$ So any initial value $(0,y_0)$ where $y_0\neq 0$ cannot be a solution of the differential equation. Also, notice that every solution of the form $y=Cx^2$ must pass through $(0,0)$, so you cannot find a unique solution at this point. In other words, if the differential equation describes the path of a particle, placing the particle initially at $(0,0)$ could lead to an infinite amount of trajectories. So the solution makes no sense at $x_0=0$.
There is no rule or convention about expressing higher order differential equations in the way you suggest. However, the Wronskian can be used to check if the general solution of a differential equation is valid for certain initial values. Your book should cover this in later chapters.