Name: Overview of Differential Equations
Uploaded: 2026-02-16T04:21:04.000Z
Duration: 14 min 4 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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.

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

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

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

That's a nice equation that we will solve.

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

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

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

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.

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

The derivative of Y, the slope depends on Y.

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

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

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

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

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

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.

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

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

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.

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

And here is a little bit more general equation.

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

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

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

That depends on the first derivative, the velocity.

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

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

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.

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

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

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

If I have a really nice function like an exponential.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

So it's first order constant coefficients.

But it's a system of N Coupled equations.

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

And we have a matrix, usually a symmetric matrix.

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

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

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.

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

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

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

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

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

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

Those steps begin with a very simple method.

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

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

Euler is fine, but not sufficiently accurate.

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

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

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