Difference between geometrical and graphical representations.

Question

My books has a sub-topic called geometric meaning of differential equations. It mentions that slope of tangent at every point in 2d space is from given differential equation. According to it, without solving the equation we can find the nature of solutions as they are family of curves passing through these points in 2d space following the direction of tangents at those points. According to my understanding what we are doing here is the graphical analysis of a differential equation. I watched 3blue1brown's video on geometric representation of derivatives where he used geometric figures and their sizes to explain the derivatives. So, please help me understand geometrical meaning of differential equations. How the slope of tangent at every point in 2d space and curves passing through them give geometric meaning of differential equaitons?

Answer 1

In 3blue1brown's video, he is looking at the change in area as you give a little increase to the sidelength of a square, thinking of the derivative the amount of area bump per sidelength bump: $\frac{dA}{dx} = \frac{d(x^2)}{dx}$. He also looked at the amount of volume bump per sidelength bump: $\frac{dV}{dx} = \frac{d(x^3)}{dx}$. He is using geometrical figures as standins for functions, and showing the derivative relationship between $x^2$ and $2x$ through their shapes.

A derivative is an operation you can do to functions. You put a function into the derivative, you get a function out. But a differential equation is asking, "What function behaves like this?" For example, "What function is twice its derivative?" would be $y = 2 * \frac{dy}{dx}$. To extend 3blue1brown's video, we might ask, "What function is $\frac{2}{x}$ times its derivative? What shape does that?" The $x^2$ square does that: $x^2 * \frac{2}{x} = 2x$

The problem here, though, is the analogy will only extend so far. Geometrical analogies for $x^2$ and $x^3$ work for degrees of 2 and 3, but don't help much at higher dimensions. And these analogies only really help in visualizing the change in $x^n$. We don't usually ask ourselves what shape might double its constant while it decreases a dimension. And that geometrical meaning of decreasing a dimension only applies to $x^n$, whereas our question might be also answered by another non-$x^n$ function. There are many other functions we want to take derivatives of and ask questions about. But if you're wanting to just get across the idea of the derivative, looking at how it works for these special geometrical $x^n$ cases can help.

Your book is taking a different approach to visualizing a differential equation. We may know how to describe a derivative as a function of x and y. We can visualize that by plotting slopes (the derivatives) on a graph and asking, "What function would act like that? What function has slopes like this at these points?" This more naturally extends into some of the physical questions that use differential equations, looking at forces in the instant to build up a more complete behavior on the whole.