Means \(order=1\), \(degree>1\). We can observe notations like \((dx)^2\), \((dy)^2\), or notations like \((dx)^p\), \((dy)^p\). And there are no notations such as \(d^2x\), \(d^2y\).
I’m running out letters for variables. I would be recycling them by personal preference.
Let \(p\) denotes \(dy/dx\) or \(y'\), and \(F_i\) be functions of x and y, \(F_i=F_i(x,y)\). The equation is solvable for p means \[(p-F_1)(p-F_i)......(p-F_n)=0\] We can see it is a factorizing polynomial of \(p=dy/dx\).
Remember \(p=dy/dx\), for an equation that we can move \(y\) to lhs, and rhs only has x and p. We can take derivative on both sides wrt x to make an order=1, degree=1 equation. Be careful when doing derivative, for example \[(px)'=(\frac{dy}{dx}x)'=p+x\frac{dp}{dx}\]
Derivative is a ratio of two differentials. we do not compute the differentials and them make the ratio. we have rules to take derivatives.
\[y = px + f(p)=Cx + f(C)\]
https://www.britannica.com/science/discriminant
We treat higher degree differential equation as a polynomial, so the discriminant gives types of the roots. The discriminant of an equation \(f(x) = 0\) can be obtained by eliminating x between \(f(x) = 0\) and \(f '(x) = 0\)
In a polynomial case, for example, \(f(x)=ax^2+bx+c=0\),
\[f'(x)=2ax+b=0 \rightarrow x=-b/2a\]
\[D=a(-b/2a)^2+b(-b/2a)+c\\=b^2/(4a)-2b^2/4a+c\\=-b^2/(4a)+c\]
The p-discriminant is the result of eliminating p between the differential equations \(f(x, y, p) = 0\) and \(\partial f(x, y, p)/\partial p = 0\).
The C-discriminant is the result of eliminating C between the solution of the differential equation \(u(x, y, C) = 0\) and \(\partial u(x, y, C)\partial C = 0\).
The differential equation, expressed in terms of the letter p. In our example, \(p=dy/dx\), \(y = px + 2p^2\) is the p-equation.
The general solution (primitive) of the differential equation. In our example, \(y = Cx + 2C^2\) is the C-equation.
In a general solution there is always a constant \(C\), varying \(C\) we can have family of curves, and just do the envelop.
https://mathworld.wolfram.com/MultipleRoot.html
https://math.stackexchange.com/questions/1858346/why-there-is-multiple-root
https://tutorial.math.lamar.edu/classes/de/RepeatedRoots.aspx
https://www.planetmath.org/discriminant
If a root of a polynomial is repeated. Discriminant vanishes we got multiple roots, discriminant does not vanish we got multiple roots. Multiple roots are not multiple roots. What is about accuracy of English?