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Linear Algebra And Differential Equations Edwards Pdf 21



OK. 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.




Linear Algebra And Differential Equations Edwards Pdf 21



So first order equations means first derivatives come into the equation. So 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 here is a nonlinear equation. The derivative of y. The slope depends on y. So it's a differential equation. But f of y could be y squared over y cubed or the sine of y or the exponential of y. So it could be not linear. Linear means that we see y by itself. Here we won't. 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. The input would change with time. Here, the input depends only on the current value of y.


So that's a big equation. 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 non-linear, 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. But that's pretty much pushing what we can hope to do explicitly and really understand the solution, because so linear with constant coefficients. Say it again. That's the good equations.


And I think of solutions in two ways. If I have a really nice function like a exponential. Exponentials are the great functions of differential equations, the great functions in this series. You'll see them over and over. Exponentials. 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.


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.


One of the options may be the favorite. ODE for ordinary differential equations 4 5. And that is numbers 4, 5. Well, Cleve Moler, who wrote the package MATLAB, is going to create a series of parallel videos explaining the steps toward numerical solution.


So that's a parallel series where you'll see the codes. This will be a chalk and blackboard series, where I'll find solutions in exponential form. And if I can, I would like to conclude the series by reaching partial differential equations.


And I would like to also include a the Laplace equation. Well, if we get there. So those are goals for the end of the series that go beyond some courses in ODEs. But the main goal here is to give you the standard clear picture of the basic differential equations that we can solve and understand.


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