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Differential Equations
A differential equation, unlike other mathematical equations, has one or more of its unknowns undergoing a continual change. These equations mathematically describe the most significant phenomena in the universe, including Newtonian and quantum mechanics, waves and oscillators, biological growth and decay, heat, economics, and general relativity. Please direct all concerns about these intricate and all-encompassing equations here.
523 Questions
Which law says If p and p q are true?
This is an incomplete statement.
Your question cannot be answered.
Why binomial distribution can be approximated by Poisson distribution?
Only in certain circumstances:
The probability of success, p, in each trial must be close to 0.
Then, for the random variable, X = number of successes in n trials, the mean is np
and the variance is np(1-p). But since p is close to 0, (1-p) is close to 1 and so np(1-p) is close to np.
That is, the mean of the distribution is close to its variance. This is a characteristic of the Poisson distribution.
Furthermore, the other characteristics of the distribution: constant probability, independence are met so the Binomial can be approximated by the Poisson.
It is possible to prove this analytically but the limitations of this browser - especially in terms of mathematical notation - preclude that.
Only in certain circumstances:
The probability of success, p, in each trial must be close to 0.
Then, for the random variable, X = number of successes in n trials, the mean is np
and the variance is np(1-p). But since p is close to 0, (1-p) is close to 1 and so np(1-p) is close to np.
That is, the mean of the distribution is close to its variance. This is a characteristic of the Poisson distribution.
Furthermore, the other characteristics of the distribution: constant probability, independence are met so the Binomial can be approximated by the Poisson.
It is possible to prove this analytically but the limitations of this browser - especially in terms of mathematical notation - preclude that.
Only in certain circumstances:
The probability of success, p, in each trial must be close to 0.
Then, for the random variable, X = number of successes in n trials, the mean is np
and the variance is np(1-p). But since p is close to 0, (1-p) is close to 1 and so np(1-p) is close to np.
That is, the mean of the distribution is close to its variance. This is a characteristic of the Poisson distribution.
Furthermore, the other characteristics of the distribution: constant probability, independence are met so the Binomial can be approximated by the Poisson.
It is possible to prove this analytically but the limitations of this browser - especially in terms of mathematical notation - preclude that.
Only in certain circumstances:
The probability of success, p, in each trial must be close to 0.
Then, for the random variable, X = number of successes in n trials, the mean is np
and the variance is np(1-p). But since p is close to 0, (1-p) is close to 1 and so np(1-p) is close to np.
That is, the mean of the distribution is close to its variance. This is a characteristic of the Poisson distribution.
Furthermore, the other characteristics of the distribution: constant probability, independence are met so the Binomial can be approximated by the Poisson.
It is possible to prove this analytically but the limitations of this browser - especially in terms of mathematical notation - preclude that.
What is pi times pi divided by pi plus pi minus pi times pi?
the same as pi squared, which is 9.86960440109
What is an example of a Rate of Change related to zoology?
The population of a species over a period of time will change according to some rate of change.
What is an example of a positive rate of change?
a car going from stoplight to next intersection accelerates at a positive rate of velocity change
How do you solve first order partial differential equation y2p - xyq x(z-2y)?
The Answers community requires more information for this question. Please edit your question using words eg "plus", "divide", "equals" because the browser used for posting questions rejects most mathematical symbols.
Simultaneous equations: x/3 -y/4 = 0 and x/2 +3y/10 = 27/5
Multiply all terms in the 1st by 12 and in the 2nd equation by 10
So: 4x -3y = 0 and 5x +3y = 54
Add both equations together: 9x = 54 => x = 6
Solutions by substitution: x = 6 and y = 8
What Differential Pressure Transmitters Do?
Differential pressure transmitters were originally designed for use in pipes to measure pressure before and after the fluid encounters a filter, pump, or another interruption in flow. Standard differential pressure transmitters come with two process connections arranged side by side to measure the drop in pressure (d) between the higher and lower points (H and L, respectively, in Figure 1). Classic differential pressure transmitters can also measure flow rates.
It wasn’t long before people realized that differential pressure measurements could be used to determine liquid level as well.
What is the derivation of Navier-Stokes equation in cylindrical coordinates for incompressible flow?
http://en.wikipedia.org/wiki/Navier-Stokes_equations Please go to this page.
How can you use the initial value theorem to solve for the initial slope of a function?
I suggest:
- Take the derivative of the function
- Find its initial value, which could be done with the initial value theorem
That value is the slope of the original function.
Where can one learn about partial differential equations?
Partial differential equations are mathematical equations that involve two or more independent variables, an unknown function, and partial derivatives of the unknown function. Even the explanation is confusing! If, however, anyone chooses to learn about PDE there are classes offered at any institution of higher learning.
Can you use differential equations in linear programming?
I'm not altogether clear about what you mean. However, the term 'linear programming' means a category of optimisation problems in which both the objective function and the constraints are linear.
Please see the link.
4.9=h-2.6
[add 2.6 to both sides]
4.9+2.6= h (+0)
h = 7.5
Who derive the schrodinger wave equation of hydrogen atom?
The Schrödinger wave equation for the hydrogen atom was derived by Austrian physicist Erwin Schrödinger in 1926 as part of his formulation of quantum mechanics. By applying his wave mechanics to the hydrogen atom, he was able to describe the behavior of electrons in terms of wave functions, which allowed for the calculation of energy levels and other properties of the atom. The derivation incorporated the Coulomb potential due to the attraction between the negatively charged electron and the positively charged nucleus. This work laid the foundation for modern quantum chemistry and atomic physics.
How is inductive reasoning used in geometry?
Inductive reasoning is used in geometry to arrive at a conclusion based on what one observes. It is not a method of valid proof, but can be used to arrive at conclusions, such as looking at a triangle with three sides and deducing that the three sides are the same based on the naked eye.
Which numerical method for solving differential equations methods gives the most inaccurate result?
Euler's Method (see related link) can diverge from the real solution if the step size is chosen badly, or for certain types of differential equations.
Are four maxwell's equations Independent?
No, Maxwell's equations are interacting partial differentials.
What is a differential equation that describe a motion of a body?
The answer depends on how the body is moving:
- in 2-dimensional space or 3-d,
- in a straight line or not
- at a constant velocity of not
- at a constant acceleration or not.
For example, the equation for a body moving in simple harmonic motion is
d2y/dx2 = -w2*x
How do you solve the time dependent Schrodinger wave equation?
If V is only a function of x, then the equation's general solution can be nearly solved using the method of separation of variables.
Starting with the time dependent Schrödinger equation:
(iℏ)∂Ψ/∂t = -(ℏ2/2m)∂2Ψ/∂x2 + VΨ, where iis the imaginary number, ℏ is Plank's constant/2π, Ψ is the wave function, m is mass, x is position, t is time, and V is the potential and is only a function of x in this case.
Now we find solutions of Ψ that are products of functions of either variable, i.e.
Ψ(x,t) = ψ(x) f(t)
Taking the first partial of Ψ in the above equation with respect to t gives:
∂Ψ/∂t = ψ df/dt
Taking the second partial of Ψ in that same equation above with respect to x gives:
∂2Ψ/∂x2 = d2ψ/dx2 f
Substituting these ordinary derivatives into the time dependent Schrödinger equation gives:
(iℏψ)df/dt = -(ℏ2/2m)d2ψ/dx2f + Vψf
Dividing through by ψf gives:
(iℏ)(1/f)df/dt = -(ℏ2/2m)(1/ψ)d2ψ/dx2 + V
This makes the left side of the equation a function of tonly, and the right a function of x only; meaning both sides have to be a constant for the equation to hold (I'm not going to prove why this is so, because if you've understood what I've written up to this point, you're probably familiar with the separation of variables technique).
That allows us to make two separate ordinary differential equations:
1) (1/f)df/dt = -(iE/ℏ)
2) -(ℏ2/2m)d2ψ/dx2 + Vψ = Eψ, where E is the constant that both separated differential equations must equal to (I did a little bit of algebra to these equations also). This is known as the time independent Schrödinger equation.
To solve 1) just multiply both sides of the equation by dt and integrate:
ln(f) = -(iEt/ℏ) + C. Exponentiating and, since C will be absorbed later, removing C:
f(t) = e-(iEt/ℏ)
Now, we have gone as far as we can until the potential, V(x), is specified. That leaves us with the general solution to the time dependent Schrödinger equation being:
Ψ(x,t) = ψ(x) e-(iEt/ℏ)
There are many solvable examples of ψ(x) for specific V(x). I've linked some below. Also, if V is a function of both x and t, there are methods to find solutions, such as perturbation theory and adiabatic approximation.
