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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 a Boussinesq approximation?
The change in density of a fluid can be neglected if the temperature change in it is small. This assumption is however not applicable when buoyant forces are in consideration as gravity changes the specific weight of a fluid considerably.
Examples of linear equations of higher order differential equation?
d2y/dx2 + 4*dy/dx + 4y = 2cos2xor
d3y/dx3 -2*d2y/dx2 + dy/dx -2y = 12*sin2x
What is oscillatory solution in differential equations?
It happens when the solution for the equation is periodic and contains oscillatory functions such as cos, sin and their combinations.
How do jet engines provide a forward thrust for an airplane by using the continuity Equation?
they move
Is vector calculus and differential equations needed for marine biology?
Marine biology is a vast discipline incorporating every aspect of biology but with a marine emphasis, this ranges from ecology (which is heavily statistical) to physiology. So in answer to your question, yes it is used, the need however depends on what path you take.
Vector calculus has proved to be useful while studying marine biology.
What is differential equation of x-y equals xy?
x - y = xy
differentiating wrt x
1 - (dy/dx) = x(dy/dx) + y
(x + 1)(dy/dx) + y + 1 = 0
What types of information can one gain from different types of maps of a country?
You can learn how to get to somewhere using road maps,museum maps,and etc.
Why you do convolution instead of multiplication?
Convolution is particularly useful in signal analysis. See related link.
What is the pendulum equation?
The period of a simple pendulum is given by the formulaT = 2*pi*sqrt(L/g)
where T = period
L = length
and g = local acceleration due to gravity.
Note that this formula is applicable only when the angular displacement of the pendulum is small. For a displacement of 22.5 degrees (a quarter of a right angle), the true period is approx 1% longer : a clock will lose more than 1/2 a minute every hour!
The only way to eliminate the arbitrary constant is if an extra equation is given that gives a value to y at a specific x.
Example:
Solve the differential equation, dy/dx = 2x + 3, where y = f(x), with condition, f(1) = 3.
Separate the variables and integrate:
dy = (2x + 3)dx, ∫ dy = ∫ (2x + 3)dx, y + C1 = x2 + 3x + C2.
C1 and C2 are arbitrary, so they combine into one constant, C:
y = x2 + 3x + C
Find C by substituting the values of the given condition into the above equation:
3 = 12 + 3(1) + C = 1 + 3 + C = 4 + C, so C = -1
Our final answer then, with the given condition, is:
y = x2 + 3x - 1
How partial differential equations are used in medicine?
Partial differential equations can be used to model physical systems over time and so can for example describe how you walk. In such an application a faulty stride can be found by comparing a patient's walk with a 'normal' walk.
What is a drawback of trying to solve a partial differential equation explicitly?
As far as I'm aware a PDE can't be solved explicitly because of the nature of the equation. However, you can separate the variables of a PDE essentially turning it into multiple ODEs which then can be solved explicitly.
The draw back of this is that most terms in a PDE are a combination of variables, such as 3xy+2yz+xyz, rather than being separated neatly already (making separation into ODEs easy), such as x+y+z. In cases where the variables are jumbled up with each other, there are often faster and easier ways of solving the PDE than trying to separate the variables into ODEs and solving explicitly.
It means that whatever you have substituted is the solution of the given linear equation. Or you have substituted the equation in itself.
How can you gain all the knowledge of the world?
Start by reading every single book, watch every movie. Then, talk to everyone in the world and learn what they know. Memorize every web site. Work every job. Travel to every location. Live forever and continue learning. Good luck.
What is 200.02 written in expanded notation?
200.02 = (2 x 100) + (0 x 10) + (0 x 1) + (0/10) + (2/100)
What is an inhomogeneous differential equation?
It is one which has a function that does not contain the dependent variable.
For example, (dy/dx) + y = f(x) is inhomogeneous
but (dy/dx) + y = 0 is not.
( f(x) is a function of the independent variable)
Who invented differential equations?
Differential equations were invented separately by Isaac Newton and Gottfried Leibniz. This debate on who was the first one to invent it was argued by both Isaac and Gottfried until their death.
What are the applications of cauchy-riemann equations in engineering?
Well, cauchy-riemann differential equation is a part of complex variables and in real-life applications such as engineering, it can be used in determining the flow of fluids, such as the flow around the pipe. In fluid mechanics, the cauchy-riemann equations are decribed by two complex variables, i.e. u and v, and if these two variables satisfy the equations in an open subset of R2, then the vector field can be asserted from the two cauchy-riemann equations, ux = vy (1) uy = - vx (2) This I think can help interpreting the potential flow (Wikipedia) in two dimensions using the cauchy-riemann equations. In fluid mechanics, the potential flow can be analyzed using the cauchy-riemann equations.
