Are all linear differential equations considered linear systems? Does the "linear" in both terms reference to the same characteristics?
http://en.wikipedia.org/wiki/Linear_system
http://en.wikipedia.org/wiki/Linear_differential_equation
Does the variable coeffcient in linear differential equations have to be polynomials? All the examples I have found seem to be polynomials for example the Cauchy–Euler equation.
For example is sin(x)y'' + cos(x^2)y' + y = 0 still considered a linear differential equation with variable...
Thanks for all your responses.
What got me to this question was how through the ages people have been racing to reach the "extremes" of mathematics and science. For example in mathematics we know pi past 2.6 trillion digits and our largest known prime number is 12,978,189 digits long. In...
Thanks.
What is the lowest uncertainty we can obtain when we measure things on the small scale, for example nanometers? For example I see people say they have made transistors at 10 nm or measured proteins at 3 nm. Is it really 10 +/- 1 nm or 10.0 +/- 0.1 nm or lower (10.000000 +/- 0.0000001...
If I understand this right, then the uncertainty in our meter measurements is:
~1/299,792,458 = ~3.33564095 × 10-9 meters or ~3 nm.
Does this mean that any measurements under 3nm will have a high level of uncertainty when we relate it back to our meter standard?
I was reading that a meter is the length of the path traveled by light in a vacuum in 1⁄299,792,458 of a second. So the speed of light is exactly 299,792,458 m/s. I am wondering what is the uncertainty in our most accurate measurement of the speed of light.
For example say I had a machine...
Nevermind, I found an answer. I was googling "big o" and could get any hits. "Computational complexity of mathematical operations" came up with better results
http://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations...
Does division take log(x) time?
For example,
if it takes ~1 millisecond to divide 7 by 4,
it will take ~2 millisecond to divide 73 by 4
it will take ~4 millisecond to divide 7334 by 4
it will take ~8 millisecond to divide 73344234 by 4
Or is it relatively constant ?
For example,
if it...
I was reading about RSA cryptography.
http://en.wikipedia.org/wiki/RSA#Key_generation
In the cryptography I start with two large prime numbers p and q. This p times q yields a large composite number n. I keep p and q secret and share n to the world. It is hard for the public to back...
So if I understand it correctly, dividing x by y computationally will take around around log(x) time.
For example,
if it takes ~1 millisecond to divide 7 by 4,
it will take ~2 millisecond to divide 73 by 4
it will take ~4 millisecond to divide 7334 by 4
it will take ~8 millisecond to divide...
I'm confused about how difficult is it to factor numbers. I am reading that it is used in encryption and it is computationally difficult, but it seems to take O(n) from how I see it.
For example to factor 6, I would
(1) divide by 2 and check if the remainder is 0
(2) divide by 3 and check...
I've taken a deeper look into this question and for a scalar Laplacian as in the heat equation, the Laplacian is the divergence of the gradient,
\nabla^2 T = \nabla \cdot \nabla T .
But for cases when the function is not a scalar such as in the Navier-Stokes Equation, the Laplacian is the...
In Batchelor's text (2000) on page 76, the stream function is defined as
\psi - \psi_0 = \int\left(u dy - v dx\right)
where \psi_0 is a constant
Now I begin with a simple function for u where
u = x^3
From mass conservation,
\frac{\partial u}{\partial x} +...
I got it working.
Using the method mentioned by HallsofIvy, I get an equation in the form
s^4 - 6s^3 + 7s^2 + 6s - 8 = 0
From this the roots, -1, 1, 2, and 4 are obtained.
This was much easier than what I was trying before. Thanks.
I am having problems solving the equation when "g(r) [...equals...] the solution to the first one"
Let X(R) equal the solution the the differential equation where G(R) = 0.
\left(\frac{d^2X}{dR^2} - \frac{2}{R^2}X\right) = G(R)
I began with the following equation...