Explanation of $n$-th order differential equation.

Question

To be more formal, an $n$th order ordinary differential equation for a function y(t) is an equation of the form $F(d^ny/dt^n , d^{n−1}y/dt^{n−1} ,...,dy/dt , y,t)= 0$. (3.4) (Of course we want $d^ny/dt^n$ to occur in $F$;if$F(¨y, ˙y, y,t) $is $y − t$ then the resulting equation $(y − t = 0)$ is not a differential equation at all.) If t does not occur explicitly in the equation, as in $dy/dt = f(y)$, then the equation is said to be autonomous.

So what does $F(d^ny/dt^n , d^n−1y/dt^n−1 ,...,dy/dt , y,t)= 0$ mean. Is $F$ a function? What does it mean since it is mentioned nowhere in book.Can someone give an intutive explanation?

Answer 1

Yes, usually, this is a function $F : \overbrace{\mathbb{R}^m \times\cdots \times \mathbb{R}^m}^{n+1 \text{ times }} \times \mathbb{R}\to \mathbb{R}^m$, or at least defined on a subset. Here, $y$ is a vector of $\mathbb{R}^m$.

Answer 2

Yes, $F$ is a function. Here's a typical example: the differential equation

$$t \dfrac{d^2 y}{dt^2} + y \dfrac{dy}{dt} + 1 = 0$$

This is $F \left(\dfrac{d^2 y}{dt^2}, \dfrac{dy}{dt}, y, t\right) = 0$ where $F(a,b,y,t) = t a + y b + 1$.