Formation of a Differential Equation

Eliminating arbitrary constants

A family of curves with $n$ arbitrary constants (like $y=Ax^2$, one constant $A$) satisfies a differential equation of order $n$, formed by differentiating the family's equation $n$ times and then eliminating the constant(s) between the original equation and its derivative(s).

Worked reasoning: one constant

For the family $y=Ax^2$ (one arbitrary constant $A$), differentiate once:

Substituting this expression for $A$ back into $y=Ax^2$:

Worked reasoning: verifying with a different one-constant family

For $y=Ae^{2x}$, differentiating gives $\frac{dy}{dx}=2Ae^{2x}=2y$ directly (since $Ae^{2x}=y$ itself) — no separate elimination step is even needed here, because the constant $A$ cancels immediately: $\frac{dy}{dx}-2y=0$.