Foundations — Classification — order, degree, linear vs nonlinear, autonomous vs non-autonomous

0. What is a function, and what is a variable?

Everything below rests on two words the parent note used without pausing: variable and function.

The picture is a single curve on a grid: horizontal axis = the input you choose, vertical axis = the output the rule returns.

Why the topic needs this: an ODE is an equation about an unknown function $y=f(x)$. If "function" and "variable" are fuzzy, every later word ("order", "linear", "autonomous") is fuzzy too.

1. The derivative $\frac{dy}{dx}$ — the star symbol

The picture: zoom into the curve until it looks like a straight line; that line's slope is $\frac{dy}{dx}$.

Why the topic needs this: the parent's very first example $\frac{dy}{dx}+y=x$ is a sentence about this slope. No derivative, no differential equation.

2. Higher derivatives — $y^{\prime \prime }$, $y^{\prime \prime \prime }$, $\frac{d^{n}y}{d{x}^n}$

The picture: the original curve bends; the amount and direction of bending is what $y^{\prime \prime }$ measures.

Why the topic needs this: order (Section 1 of the parent) is nothing but "the largest $n$ that appears in $y^{(n)}$".

3. Powers of derivatives — $(y^{\prime }{)}^2$, ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3$

Why the topic needs this: degree = the power on the highest derivative. To find degree you must be able to see a power like the $3$ in ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3$ and know it is separate from the "twice-differentiated" written inside.

4. Polynomials, radicals, and "transcendental" functions

The parent's degree rule says "make it a polynomial in the derivatives, clearing radicals", and its linear rule bans "transcendental functions of $y$". Three vocabulary items:

Why the topic needs this: these three words are the exact gatekeepers of the degree test and rule 3 of linearity.

5. Coefficients $a_k(x)$ and forcing $g(x)$

The parent writes the general linear ODE as $a_n(x)y^{(n)}+\cdots +a_0(x)y=g(x)$.

Why the topic needs this: without the idea of "coefficient of $y$" versus "the $y$ itself" you cannot state the linear form at all.

6. The slope field — the picture behind autonomy

The picture below shows the payoff the parent claims for autonomous equations $y^{\prime }=f(y)$: since $f$ ignores $x$, the arrows are identical along every vertical $x$-line, so any solution slid left/right is still a solution.

Why the topic needs this: "autonomous = the clock is not watched" is a visual fact about this field — the arrows don't change as $x$ advances. That is the doorway to the Phase Line and Equilibria.

7. Putting the four questions in order

Each foundation feeds exactly one of the parent's four checks, and all four merge into the single act of classification. From there the road forks toward Separable Equations, Linear First-Order ODEs and Integrating Factors, Second-Order Linear ODEs with Constant Coefficients, the Superposition Principle, and Existence and Uniqueness (Picard–Lindelöf) — every one of which first assumes you can classify.

Equipment checklist

Read each line, cover the right side, and check you can answer before revealing.