Homework Statement
Find all solutions of the IVP y'' + a(t)y' + b(t)y = 0, y(t0) = 0, y'(t0) = 0 where t0 is any fixed point on the t-axis and the coefficients are continuous.
The Attempt at a Solution
I know this has to do with the Existence and Uniqueness theorem. How would I apply...
Homework Statement
We have the ODE y' = -ky + R for a population y(t) where death rate exceeds birth rate, counteracted by a constant restocking rate.
I'm assuming k is the decay constant and R is the restocking rate
The population at time t0 = 0 is y0, and I have to find a formula for...
so what are some simple unsolved problems in astrophysics, that are interesting. could someone just list some topics out? I've been researching on the interwebz, but can't seem to find a topic that is interesting/simple.
also are there any such topics related to newtonian mechanics/electricity...
So I know that quandles are associated with knot theory, etc. but what are "shadow colorings" ?
on a broader context, can someone plz give me a simple definition of a "knot invariant" and a "quandle" ?
and what does this have to do with "racks" ?
sorry, all this esoteric language is...
Homework Statement
Newspaper Boat- So I have been thinking about designing a newspaper boat out of pure entertainment. I want to build a boat out of shellack, tape, and newspaper, to maximize speed (to race people in swimming pools). considering i have some amount of these supplies, what...
Ok I feel this is the best place to place this subject. Well, I have been interested in materials science and crystallography lately, especially the future applications of this. What are some future materials we could make from studying crystal growth? How could we go about accomplishing this...
Homework Statement
So does anyone here know a lot about ballistic pendulums? I was just wondering how I could calculate the distance the ball will travel horizontally when shot horizontally, from a table.
Homework Equations
The Attempt at a Solution
I know that first, I have to...
Hey, does anyone know much about knot invariants and the stuff? Including Fox Calculus, Alexander Polynomials, Reidemeister theory? Any helpful sites? Tips for learning about this subject? Also, what are quandles and how are they related? Thanks!
Homework Statement
What is the line integral of F(x,y,z) = (xy, x, xyz) over the unit circle c(t) = (cost, sint) t E (0,2pi) ?
Homework Equations
integral= (f(c(t))*c'(t))dt)
The Attempt at a Solution
Ok, so I tried solving this like I would any other line integral using the given...
Homework Statement
The question is: Romeo and Juliet are dating on a canoe in a lake. They are 4 feet apart, and Romeo wants to see how much Juliet weighs. He is on the left, Juliet on the right. So, he decides to switch spots with her. After switching, the boat moves 0.444 feet to the left...
Homework Statement
So, this problem is about energy. The question is: what must the minimum height (in R) of a height be, for a ball to go down it, through a loop the loop (with radius R), and then through a rough surface 6R's long with u= 0.5.
Homework Equations
gravitational potential...
Homework Statement
Hey guys, I have one question: how can I integrate the function f(x,y,z)=x + y + z over the region between the paraboloid 4-x^2-y^2 and the xy-plane?
Homework Equations
For the paraboloid region, I used polar coordinates and found the volume of the region to be 8pi...
Homework Statement
What is the x-simple description of a 4-ball = { ||x|| \leq r }
Homework Equations
It's a 4-ball, so isn't the equation x^{2} + y^2 +z^2 +w^2 ?
The Attempt at a Solution
For my limits, I got x is between \pm \sqrt{1-z^2} and y is between -1 and 1, and z is...
Homework Statement
So my question is: what is the volume of the region R between the paraboloid 4-x^2-y^2 and the xy-plane?
Homework Equations
I know how to solve it, it is a triple integral, but how do you find the limits of integration?
The Attempt at a Solution
Do I set x=0...
How do you solve the double integral of xe^(xy)? bounded by x=0, y=1, x^2-y=0. I used both x-simple and y-simple methods but neither worked..I don't know what the limits are...thanks!
Homework Statement
Find the equation of the plane through vectors u=(1,1,2) parallel to the plane containing v=(2,2,-1), w=(1,-1,0), and the origin.
Homework Equations
The Attempt at a Solution
I did the cross product of v x w = (-1,-1,-4).
Thus, my equation became -x-y-4z=-9...