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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
What is the general solution of a differential equation?
It is the solution of a differential equation without there being any restrictions on the variables (No boundary conditions are given). Presence of arbitrary constants indicates a general solution, the number of arbitrary constants depending on the order of the differential equation.
How do you figure goal differential?
Goal differential is calculated by subtracting the total number of goals conceded from the total number of goals scored by a team over a specified period, such as a season. The formula is: Goal Differential = Goals Scored - Goals Conceded. A positive goal differential indicates a team has scored more goals than it has allowed, while a negative differential shows the opposite. This statistic is often used to assess a team's overall performance in sports like soccer or hockey.
What are the applications of cauchy-riemann equations?
The Cauchy-Riemann equations are fundamental in complex analysis, providing conditions for a function to be holomorphic, meaning it is complex differentiable. These equations are essential in various fields, including fluid dynamics, where they describe potential flow, and in electrical engineering for analyzing electromagnetic fields. Additionally, they are used in conformal mapping, which allows for the transformation of complex shapes in a way that preserves angles, facilitating the solution of physical problems in engineering and physics.
Why does charge developed by induction disappear quickly?
"Charge?"
The field produced by an inductor exists ONLY while the current flow is changing,
What is the derivative of cos pi x plus sin pi y all to the 8th power equals 44?
(cos(pi x) + sin(pi y) )^8 = 44
differentiate both sides with respect to x
8 ( cos(pi x) + sin (pi y ) )^7 d/dx ( cos(pi x) + sin (pi y) = 0
8 ( cos(pi x) + sin (pi y ) )^7 (-sin (pi x) pi + cos (pi y) pi dy/dx ) = 0
8 ( cos(pi x) + sin (pi y ) )^7 (pi cos(pi y) dy/dx - pi sin (pi x) ) = 0
cos(pi y) dy/dx - pi sin(pi x) = 0
cos(pi y) dy/dx = sin(pi x)
dy/dx = sin (pi x) / cos(pi y)
What is potential differential and why is it important?
Potential differential, often referred to as the electrochemical potential difference, is the difference in electric potential between two points, which drives the movement of charged particles, such as ions, across a membrane. It is crucial in biological systems, particularly in neurons and muscle cells, as it governs processes like action potentials and synaptic transmission. Understanding potential differentials is essential for studying cellular communication, metabolism, and overall physiological functions. Additionally, it has applications in fields like bioengineering and pharmacology, influencing drug delivery and the development of medical devices.
What does it mean when a differential equation is linear?
It means that the dependent variable and all its derivatives are multiplied by constants only, not by themselves nor by functions containing the independent variable..
For example, (dy/dx) + xy = 0 is non-linear
but (dy/dx) + y = (x^2)coswx is linear.
(Note that it doesnt matter how the function of the independent variable is)
In a Solow model, a differential equation exists because the optimal growth rate is a difference between two functions, whose optimisation is their derivative set equal to zero. Consider:
Break-even investment is equivalent to the minimal level to maintain the capital-labour ratio:
(n + g + d)k(t)
And actual investment is:
sf(k(t))
The differential solution to this equation describes the optimal outcome. Specifically, we optimise economic growth by choosing the savings versus consumption ratio such that the equation
sf(k(t)) - (n + g + d)k(t)
is optimised. This equation represents the derivative of the capital-labour ratio. Therefore, its optimisation is equivalent to
0 = sf(k(t)) - (n + g + d)k(t)
thus
sf(k(t)) = (n + g + d)k(t)
when k(t) = f(k(t)), then
s = n + g + d
What is the heat loss equation?
Heat loss due to change in temperature:
Q = mc(T2-T1)
Heat loss due to change in phase:
Q = mL
c and L are constants that are specific to each compound at certain temperatures. For water, we usually take c to be 4186 J/(kg*K).
Busbar differential protection is a safety mechanism used in electrical power systems to detect faults specifically within busbars, which are conductive bars that distribute electrical power. It compares the incoming and outgoing current to identify any discrepancies that indicate a fault, such as short-circuits or equipment failures. If the difference exceeds a predetermined threshold, the protection system triggers a circuit breaker to isolate the affected area, ensuring system stability and safety. This type of protection is crucial for preventing equipment damage and maintaining the reliability of power distribution networks.
What is the least number of moves in the Tower of Hanoi puzzle with only 5 disks?
To move n disks, you need 2n-1moves. In this case, 31.
What is 7 over 9 plus 5 over 9 equals?
To word out an improper fraction if it was the same denominator, yes, you just add the numerator and it equals 12/9, but as a mixed number it equals 1 1/3 (one and one third). It depends which one you want. :)
What is the importance of boundary conditions in solving the schrodinger's wave equation?
Boundary conditions allow to determine constants involved in the equation. They are basically the same thing as initial conditions in Newton's mechanics (actually they are initial conditions).
