Ordinary Differential Equations class

[Jonnyb42]

I am a freshman in college and I am in multivariable calculus, I took the first AP calc in high school. I am deciding to walk in on an Ordinary Differential Equations class. It is apparently graduate and I didn't understand a single thing in there. I didn't understand anything mainly because I do not know what the symbols represent, and the professor was impossible to understand, (old guy.) Anyways, I took the notes down verbatim and I was wondering if some of you could explain to me what the different sybmols meant...
I understand it is a huge jump from multivariable calculus to ODE, so please no negative comments.

One of the rather longer equations. I believe it was part of a theorem.
[PLAIN]http://mynqa.com/Cargo/eqn.jpg
(sorry if it is hard to read, I could not manage the latex form, I will have to learn how to use that sometime)

what is the function $$\varphi$$(t,$$T$$,$$\varsigma$$) for example?
I also do not understand the inputs to f, what does this equation mean in general?

If it is too difficult to explain in one response, I am also fine with links to websites that could explain, or even reference a book.

 

[Eynstone]

This is an integral equation. It's not possible to make out heaad or tail out of it unless one knows the context.

 

[HallsofIvy]

I am a freshman in college and I am in multivariable calculus, I took the first AP calc in high school. I am deciding to walk in on an Ordinary Differential Equations class. It is apparently graduate and I didn't understand a single thing in there. I didn't understand anything mainly because I do not know what the symbols represent, and the professor was impossible to understand, (old guy.) Anyways, I took the notes down verbatim and I was wondering if some of you could explain to me what the different sybmols meant...
I understand it is a huge jump from multivariable calculus to ODE, so please no negative comments.

One of the rather longer equations. I believe it was part of a theorem.
[PLAIN]http://mynqa.com/Cargo/eqn.jpg
(sorry if it is hard to read, I could not manage the latex form, I will have to learn how to use that sometime)

what is the function $$\varphi$$(t,$$T$$,$$\varsigma$$) for example?

It is the unknown function the equation is describing.

I also do not understand the inputs to f, what does this equation mean in general?

f is the function that essentially determines the equation. Perhaps it would make more sense if this were written in Roman letters rather than Greek:
$$y(t_0, t_1, x)= x+ \int_{t_0}^{t_1} f(s, y) ds$$
That is, if you take this unknown function y, put it into the given function f as the second variable and integrate f(s, y) dx from $t_0$ to $t_1$ and, finally, add x, you get back the function y.

This is, as eynstone said, an "integral equation". Rather than giving information about the derivatives of the unknown function, as a "differential equation" would, it gives information about the integral of the function.

If it is too difficult to explain in one response, I am also fine with links to websites that could explain, or even reference a book.

 

[Jonnyb42]

Thank you, using roman letters does make it make way more sense, I am not sure why the professor has to use greek letters, it's fine anyways.

So when you say it is the unkown function, you are referring to the primary function in an ODE? Is this because in an ODE there is only one function that has derivatives in the same equation? such as y'+5 = 3y

Is the "y" in my example equation above the same y you have in your roman equivalent equation?

One more thing, the ends of your integration are backwards, in my notes it is written from T to t over and over again, and I know I didn't copy it wrong, I am confused though why he has it seemingly backwards like that.

This is beginning to make more sense, today's lecture made more sense than the last. Thank you for helping me.

 

[jackmell]

I am a freshman in college and I am in multivariable calculus, I took the first AP calc in high school. I am deciding to walk in on an Ordinary Differential Equations class. It is apparently graduate and I didn't understand a single thing in there. I didn't understand anything mainly because I do not know what the symbols represent, and the professor was impossible to understand, (old guy.) Anyways, I took the notes down verbatim and I was wondering if some of you could explain to me what the different sybmols meant...
I understand it is a huge jump from multivariable calculus to ODE, so please no negative comments.

One of the rather longer equations. I believe it was part of a theorem.
[PLAIN]http://mynqa.com/Cargo/eqn.jpg
(sorry if it is hard to read, I could not manage the latex form, I will have to learn how to use that sometime)

what is the function $$\varphi$$(t,$$T$$,$$\varsigma$$) for example?
I also do not understand the inputs to f, what does this equation mean in general?

If it is too difficult to explain in one response, I am also fine with links to websites that could explain, or even reference a book.

That's a mess dude. First thing is that it's apparently a graduate-level class. You need to walk in an undergraduate ODE class where they study intro ODEs that are much easier to understand and follow. You know, separation of variables, first order ODEs, linear DEs and the like. In an intro ODE class, the only integral equations you'll encounter is in the second semester involving simple integral equations involving convolutions and not the likes above.

It's a simple equation for me: get out of that class and find a first semester ODE class to audit.

 

[Jonnyb42]

I know that a undergraduate level class would be more appropriate, however there are none currently that fit into my schedule :(

 

[n1person]

haha, that is sort of a funny reason to take a class.

It is pretty much equivalent to saying "Oh, I cannot fit in Calculus into my schedule, so I suppose I will take Algebraic Topology," or, "Oh I cannot take my Intro Physics class due to a course conflict so I will take Quantum Field Theory."

Simply put, if you don't have the pre-reqs don't take the course! If it is a graduate class bets are it is pretty much completely proof based? Do you know how to write analysis proofs? If you only had AP Calculus I can say with much certainty that you don't know how.

If the first day of class they use completely foreign notation then bets are you should drop the class like its hot!

Also it should be noted that it is not nessisarily a "huge jump" from Multi to ODE. For lots of ODE stuff you only need multi for theory, and even then it is usually just partial derivatives (exact equations, linearization, uniqueness of solutions, etc), but it is a huge jump from multi a graduate ODE class. Usually at the graduate level they don't even solve ODEs! It is all uniqueness and existence of solutions and stuff like that.