Name: 微分方程大解密！ (Overview of Differential Equations)
Uploaded: 2026-02-16T04:24:56.000Z
Duration: 14 min 4 s

Okay, well the idea of this first video is to tell you what's coming to give a kind of outline of what is reasonable to learn about ordinary differential equations.

好的，這個第一個影片的宗旨是告訴大家接下來會有什麼內容，大致介紹一下關於常微分方程，有哪些東西是值得學習的。

And a big part of the series will be videos on first order equations and videos on second order equations.

這個系列影片中，很大一部分會是關於一階方程的影片，以及關於二階方程的影片。

Those are the ones you see most in applications and those are the ones you can understand and solve when you're fortunate.

這些是在實際應用中你最常看到的，也是你幸運時能夠理解和求解的。

So first order equations means first derivatives come into the equation.

所以一階方程的意思是，方程裡會出現一階導數。

That's a nice equation that we will solve.

這是一個我們將會求解的漂亮方程。

We'll spend a lot of time on the derivative is that's the rate of change of Y.

我們會花很多時間在導數上，因為那是 Y 的變化率。

The changes in the unknown Y as time goes forward are partly from depending on the solution itself.

隨著時間的推移，未知數 Y 的變化，一部分是來自它本身的解。

That's the idea of a differential equation, that it connects the changes with the function Y as it is, and

這就是微分方程的概念，它連接了變化與函數 Y 本身，以及

then you have it inputs called Q of T which produce their own change, they go into the system, they become part of Y and they grow, decay, oscillate, whatever Y of T does.

然後你會有一個叫做 Q of T 的輸入，它們會產生它們自己的變化，它們進入系統，成為 Y 的一部分，然後它們成長、衰減、振盪，無論 Y of T 做什麼。

So that is a linear equation with a right hand side with an input, a forcing term.

所以這是一個線性方程，右邊有一個輸入，一個強迫項。

The derivative of Y, the slope depends on Y.

Y 的導數，也就是斜率，取決於 Y。

But F of Y could be Y squared or Y cubed or the sine of Y or the exponential of Y.

但 F of Y 可能會是 Y 平方、Y 立方、Y 的正弦函數，或是 Y 的指數函數。

線性代表我們看到的是單獨的 Y。

Well, we'll come pretty close to getting a solution because it's a first order equation and the most general first order equation, the function would depend on T and Y.

嗯，我們會很接近得到一個解，因為這是一階方程，而最一般的一階方程，函數會取決於 T 和 Y。

Here the input depends only on the current value of Y.

在這裡，輸入只取決於 Y 的目前值。

I might think of Y as money in a bank, growing, decaying, oscillating.

我可能會把 Y 想成銀行裡的錢，它會成長、衰減、震盪。

Or I might think of Y as a distance on a spring.

或者我會把 Y 想成彈簧上的距離。

Ok, so those are first order equations and second order have second derivatives.

好的，所以那些是一階方程，而二階方程有二階導數。

The second derivative is the acceleration.

It tells you about the bending of the curve.

If I have a graph, the first derivative we know gives the slope of the graph.

如果我有一個圖形，我們知道一階導數代表圖形的斜率。

Is it going up, is it going down, is it a maximum?

它是在上升、下降，還是極大值？

The second derivative tells you the bending of the graph, how it goes away from a straight line and that's acceleration.

二階導數告訴我們圖形的彎曲，它如何偏離直線，這就是加速度。

So Newton's law, the physics we all live with would be acceleration is some force and there is a force that depends again linearly.

所以牛頓定律，我們都生活的物理定律，就是加速度等於某個力，而這個力又線性地取決於。

That's a key word on Y, just Y to the first power.

那是 Y 的一個關鍵詞，就是 Y 的一次方。

And here is a little bit more general equation.

In Newton's law, the acceleration is multiplied by the mass.

在牛頓定律裡，加速度會乘以質量。

So this includes a physical constant here, the mass Then there could be some damping.

所以這裡包含了一個物理常數，質量，然後可能會有某種阻尼。

If I have motion, there may be friction slowing it down.

如果我有運動，可能會因為摩擦力而減速。

That depends on the first derivative, the velocity.

這取決於一階導數，也就是速度。

And then there could be the same kind of force term that depends on Y itself.

然後可能會有一個與 Y 本身有關的相同類型的力項。

And there could be some outside force, some person or machine that's make force creating movement, an external forcing term.

也可能會有一個外部的力，像是某個人或機器產生的力，這是一個外部強迫項。

And let me just say at this point we let things be nonlinear and we had a pretty good chance if we get these to be nonlinear, the chance at second order has dropped.

在這一點上，我只想說，我們讓這些東西是非線性的，而且如果我們讓這些變成非線性的，二階方程的機率就降低了。

And the further we go, the more we need linearity and maybe even constant coefficients M, B and K.

而且我們走得越遠，就越需要線性，甚至可能需要常數係數 M、B 和 K。

So that's really the problem that we can solve as we get good at it is a linear equation, second order, let's say with constant coefficients.

所以這真的是我們在擅長之後可以解決的問題，也就是線性方程，二階的，假設是常數係數。

That's pretty much pushing what we can hope to do explicitly and really understand the solution.

這幾乎是我們希望明確做到並真正理解解的極限了。

That's the good equations and I think of solutions in two ways.

這些是好的方程，我從兩個方面來思考解。

If I have a really nice function like an exponential.

如果我有一個很棒的函數，像是指數函數。

Exponentials are the great functions of differential equations, the great functions in this series.

指數函數是微分方程的絕佳函數，是這個系列中的絕佳函數。

Say F of T equals E to the T or E to the Omega T or E to the I omega T that I is the square root of minus 1.

說 F of T 等於 E 的 T 次方，或是 E 的 Omega T 次方，或是 E 的 i Omega T 次方，這裡的 i 是負一的平方根。

In those cases we will get a similarly nice function for the solution.

在這些情況下，我們會得到一個同樣不錯的函數作為解。

We get a function that we know, like exponentials, and we get solutions that we know.

我們得到一個我們認識的函數，像是指數函數，也得到我們認識的解。

Second best are we get some function who we don't specially know.

其次是我們得到一個我們不太認識的函數。

In that case, the solution probably involves an integral F or 2 integrals of F.

在那種情況下，解可能包含一個 F 的積分，或是 F 的兩個積分。

That formula includes an integration that we would have to do.

這個公式包含一個我們必須做的積分。

And then when we get to completely nonlinear functions, or we have varying coefficients, then we're going to go numerically.

然後當我們遇到完全非線性函數，或是係數會變動時，我們就會用數值方法。

So really the wide, wide part of the subject ends up as numerical solutions.

所以這個主題大部分的內容，最後都會變成數值解。

But you've got a whole bunch of videos coming that have nice functions and nice solutions.

不過接下來會有好多影片，會教大家處理有漂亮函數和漂亮解的狀況。

Ok, so that's first order and second order.

好的，所以這是一階和二階的介紹。

Because a system doesn't usually consist of just a single resistor, a single spring.

因為一個系統通常不只包含一個電阻、一個彈簧。

In reality we have many equations and we need to deal with those.

實際上我們有很多方程需要處理。

So Y is Now a vector y1, y2 to yn n different unknowns, N different equations.

所以 Y 現在是一個向量，y1, y2 到 yn，代表 n 個不同的未知數，以及 N 個不同的方程。

所以這裡是一個 n 乘 n 的矩陣。

So it's first order constant coefficients.

But it's a system of N Coupled equations.

但這是一個 N 個耦合方程的系統。

And so is this one with the second derivative, second derivative of the solution, but

這個也是一樣，有二階導數、解的二階導數，但是

And we have a matrix, usually a symmetric matrix.

我們有一個矩陣，通常是對稱矩陣。

So again, linear constant coefficients, but several equations at once.

所以，又是線性常係數，但同時處理好幾個方程。

And that will bring in the idea of eigenvalues and eigenvectors.

這將會引出特徵值和特徵向量的概念。

Eigenvalues and eigenvectors is a key bit of linear algebra that makes these problems simple because it turns this coupled problem into N uncoupled problems, N first order equations that we can solve separately or n second order equations that we can solve separately.

特徵值和特徵向量是線性代數中一個很重要的部分，它能簡化這些問題，因為它能將耦合問題轉化為 N 個獨立的問題，N 個一階方程，我們可以分開解，或者 N 個二階方程，我們也可以分開解。

That's the goal with matrices, is to uncouple them.

使用矩陣的目標就是將它們解耦。

And then really the big reality of this subject is that solutions are found numerically and very efficiently.

然後這個主題的真正重點是，解都是用數值方法找到的，而且非常有效率。

And there's a lot to learn about that, a lot to learn.

關於這點，有很多東西可以學，真的有很多東西可以學。

And Matlab is a first class package that gives you numerical solutions with many options.

而 Matlab 是一個一流的套件，能提供你很多選項的數值解。

One of the options may be the favorite ODE for ordinary differential equations.

其中一個選項可能是常微分方程裡最受歡迎的 ODE。

Well, Cleave Molar, who first wrote the package MATLAB is going to create a series of parallel videos explaining the steps toward numerical solution.

嗯，Cleve Moler，也就是最先寫 Matlab 套件的人，他將會製作一系列的平行影片，來解釋數值解的步驟。

Those steps begin with a very simple method.

這些步驟從一個非常簡單的方法開始。

Maybe I'll put the creator's name down Euler.

或許我會把創作者的名字寫成 Euler。

So you can know that because Euler was centuries ago, he didn't have a computer, but

所以你知道，因為 Euler 是幾個世紀前的人，他沒有電腦，但是

所以 Euler 可能是 ODE 的第一種方法。

Euler is fine, but not sufficiently accurate.

That 4 and 5 indicate a much higher accuracy, much more flexibility in that package.

那個 4 和 5 表示套件中有更高的精確度和更大的彈性。

So starting with Euler, Cleve Moller will explain several steps that reach a really workhorse package.

所以從 Euler 開始，Cleve Moler 會解釋幾個步驟，最終達到一個真正實用的套件。

So that's a parallel series where you'll see the codes.

所以那是一個平行系列，你會看到程式碼。