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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 number is a solution of the inequality 10.6 b?
The answer to this question is 14. The reason why is becasue 14 is greater than 14
Which law says If p and p q are true?
This is an incomplete statement.
Your question cannot be answered.
How do you convert second order differential equations to first order differential equations?
You can't convert a second order DE to first order except in special cases (like an ODE with y'' and y' but no y terms).
HOWEVER, you can convert a second order ODE into a systemof first order ODEs:
Assume it is of the form f(x, y, y', y'') = 0, where y(x) is the solution.
1) Let u1 = y and u2 = y'
2) Substitute y'' for u2', y' for u2, and y for u1 to get eq1
3) u2 = u1' is eq2.
eq1 and eq2 are a system of two first-order ODEs which represent the same problem.
What is IVP in differential equations?
Initial Value Problem.
A differential equation, coupled with enough initial conditions for there to be a unique solution.
Example:
y'' - 6y = exp(x) ; y'(0) = y(0) = 0
An irreducible equation is an irreducible polynomial which is equal to zero. A polynomial is irreducible over a particular type of number if it cannot be factorised into the products of two or more lower degree polynomials with coefficients of that type of number. For example, the equation x2 + 1 =
0 is irreducible over the real numbers; there are no lower order polynomials, containing only real coefficients, which could be multiplied together to give this equation.
Why is finding mean median and mode necessary?
They help to analyse data samples and therefore compare statistics. Normally all three methods are used as they all contain weaknesses. However the mean is usually the normal statistic used after anomalous results have been removed
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.
Is speech is differential equation?
Speech itself is not a differential equation, but the processes involved in speech production and perception can be modeled using differential equations. For instance, the mechanics of airflow and vocal cord vibrations can be described mathematically with differential equations to simulate sound wave propagation. Additionally, models of auditory processing in the brain may also utilize differential equations to represent changes over time in response to speech signals.
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.
