This Week's Insights into Differential Equations
This week I worked through two differential equation problems: the "missile tracking" problem and the "solar furnace" problem. In the process I deepened my understanding of differential equations and became more familiar with the relevant manipulations. I'm recording this here mainly to invite discussion.
I. The Nature of Differential Equations
Many readers know that ever since Newton and Leibniz invented calculus, it has rapidly permeated almost every discipline, later giving rise to a number of remarkable branches such as the calculus of variations and differential equations. It is well known that differential equations are a tool for solving many important problems. I wonder how well readers understand differentials and differential equations. Actually, for ordinary differential equations, their essential nature is the same as that of the algebraic equations we've already learned — the only difference is that, besides the usual operations of addition, subtraction, multiplication, division, and exponentiation, there are two additional relations at play: differentiation and integration. For instance, for the first-order differential equation $\dot{y}=f(x,y)$, one might think of it as an equation in two variables, but actually it isn't — this is a system consisting of "four unknowns and three equations." We can write it as
$$dy=f(x,y)dx,y=\int dy,x=\int dx$$
where the unknowns are respectively $x,y,dx,dy$. Here it's worth noting one point: although dx and dy tend to zero, they are variables, not fixed constants (please consult a book on rigorous calculus analysis for the relevant details). You might wonder what benefit there is in describing things this way. In fact, this helps reduce the sense of mystery surrounding differential equations (note: not the difficulty), while also helping guide us in a directed way when eliminating unknowns in a system of differential equations. Once we understand this, we can deliberately use ordinary algebraic operations together with differentiation to eliminate unknowns, leaving only x and dx (or y and dy).
II. Computing Differentials
This week I explored the relationship between $\frac{d^2 y}{dx^2}$ and $\frac{d^2 x}{dy^2}$. Let $\frac{dy}{dx}=P$, then $\frac{d^2 x}{dy^2}=\frac{d(1/P)}{dy}=-\frac{dP}{P^2 dy}=-\frac{dP dx}{P^2 dx dy}$
where $\frac{dP}{dx}=\frac{d^2 y}{dx^2}$, so $\frac{d^2 x}{dy^2}=-\frac{\frac{d^2 y}{dx^2}}{(\frac{dy}{dx})^3}$. After simplification we find
$$\frac{dy}{dx}=-\frac{d^2 y}{d^2 x}$$
Readers might feel this is a rather nice conclusion, linking higher-order differentials to first-order differentials. However, although this conclusion isn't wrong, it is imprecise, even meaningless. The reason is that in dealing with differentials I overlooked an important issue: in general, the $d^2 y$ in $\frac{d^2 y}{dx^2}$ is not equal to the $d^2 y$ in $\frac{d^2 y}{dt^2}$. That is to say, even for one and the same function, its second-order differential is not fixed — it depends on the choice of independent variable. So what meaning does the expression above actually have?
——————————— BoJone, 2010.11.03
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.