\nabla^2 \phi(x,y) = - \frac{k} {I_0} \frac {dI(x,y)} {dz}
hmm yeah, sorry, I really have no idea what you are talking about but thought I would clean it up a bit. :-p
What is z? the z axis? or the xy plane?
Series solution of 2nd Order Linear Equations: Ordinary Point
Awsome, I feel special now that this was made a sticky
I haven't been around in a while, I was really busy during finals and then kinda crawled in a hole for a month during break. but for now I am back, we will see how long it...
Nonhomogeneous Equations; Variation of Parameters
We have already seen how to find a particular solution for nonhomogeneous equation using the Method of Undetermined Coefficients, now we will try to use variation of parameters to accomplish the same thing.
Let's jump straight to an example...
Nonhomogeneous Equations; Method of Undetermined Coefficients
Here we will look at Second Order, nonhomogeneous Linear Equations of the form;
L[y] = y'' + p(t)y' + q(t)y = g(t)
where p(t), q(t), and g(t) are continuous functions on an open interval, I. We can use the homogeneous...
Are you talking about the sums in series solutsions of 2nd Order Linear Equations? If it is beyond that, sorry I can't help ya. I'm only taking this course just now.
Repeated Roots; Reduction of Order
This is a method for finding a second solution to a 2nd order linear homogeneous differential equations with constant coefficients assuming you already have the first solution. It can often occur when the roots of the equation are the same (when b2 - 4ac = 0)...
Complex Roots; 2nd Order Homogeneous Constant Coeff. Diff. Eqs.
I have fallen way behind in doing this, about 3 + 3 fairly complicated sections, so I will try to cover as much as I can during my time off tonight and tomorrow morning.
So, we already saw what happens to 2nd order linear...
I'm actually not entirely sure how to solve this. This is not something we covered (higher order linear equations)
Skimming through the section, it looks like it is done in the same way as regular constant coefficient homogeneous equations.
3r3 - 2r2 + 1r = 0
where r are the kernals...
Second Order Linear Differential Equations
A second order differential equation is linear if it is in the following form;
y'' + p(t)y' + q(t)y = g(t)
or
P(t)y'' + Q(t)y' + R(t)y = g(t)
Where p, q, and g and functions of t
If the problem has initial conditions it will be in the form of...
OD Exact Equations and Integrating Factors
Theorem:
Let the functions M, N, My, Nx, where subscripts denote partial derivatives, be continuous in the rectangular region R: [alpha] < [beta], Γ < y < δ. Then;
M(x,y) + N(x,y)y' = 0
is an exact differential equation in R if and...
I guess I am getting confused because the book does not use
M = max |f(t,y)|
C = min (A,B/M)
notation. From what I understand, all this really says is that you are trying to find the maximum interval of f(t,y), where the limiting factor is either A or B/M (which is denoted C). But why B/M...
Interval of Solution for Nonlinear First Order Equations
I will mention right now, I have some difficulty understanding this part so I will have even more difficulty explaining it and will probably need a little help. The good news, this isn't the most important topic in my opinion.
The...
"I've often wondered why this is the case. dy and dx are variables right? So why can't they be treated as such?"
x is a variable, y is a funtion of x. By cross multiplying you would be treating y like a variable.
And your answer is correct.
Separable Equations
I'm terribly sorry, I have been entirely too busy recently but I'm back, for the moment.
So now that we know how to differentiate a first order linear equation with variable coefficients, let us move on to linear separable equations.
In the last section we used the...
Yes, your answer is correct. It actually took me a while to get it. Out of curiousity, why did you change to using the variable s? Are you just used to using it and forgot that it was in terms of t or can this be done?
Thank you for your participation. Your solution is infact correct except for the constant is missing. No biggy, I always forget those too. Did you find it hard to follow without the book and should I have presented this more clearly some how?
I'm glad to know that someone else knows this...
First Order Differential Equations
"This chapter deals with differential equations of the first order
\frac {dy} {dt} = f(t,y)
where f is a given function of two variables. Any differentiable function y = Φ(t) that satisfies this equation for all t in some interval is called a...
My intent is to create a thread for people interested in Differential Equations. However, I will explicitly state that I am only a student of this class myself and that many things could end up being incorrect or an improper way to present the material.
I will merely be going along with the...