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.

975 Questions
Math and Arithmetic
Units of Measure
Differential Equations

What does the '' and ' mean in length measurements?

One ' (an apostrophe) means feet and 2 " (quotation marks) means inches.

Building and Carpentry
Weight and Mass
Differential Equations

What is the weight of a red clay brick in Kilograms?

The red clay bricks I bought today - measuring about 4" x 8" x 2 1/4" thick (10 cm x 20 cm x 5 1/2 cm) - weigh just under 6 lbs (about 2.7 Kilograms) apiece.

Math and Arithmetic
Differential Equations

Find the second derivative for the square root of x2 - 3?

f(x) = x2 - 3

By the power rule:

f'(x) = 2x2-1 - 0

= 2x

f''(x) = 2(1)x1-1

= 2

The second derivative of x2 - 3 is 2.

Math and Arithmetic
Differential Equations

What is the difference between an ordinary differential equation and a partial differential equation?

ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.

Differential Equations

1 katha how much decimal?

Here we go....... 1 Bigha = 20 katha = 33 decimal 1 katha = 1.65 decimal

Differential Equations

If Shawn uses a 45N of force to stop the cart 1 meter from running his foot over How much work does he do?

you times 45 x 1 = 45

Differential Equations

What are the physical significance of Maxwell's equations?

As per my knowledge,Maxwell's equations describes the relations between changing electric and magnetic fields. That means time varying electric field can be produced by time varying magnetic field and time varying magnetic field can be produced by time varying electric field.

Math and Arithmetic
Prefixes Suffixes and Root Words
Differential Equations

What is after a quadrillion?

Billion has 9 zeros

Trillion has 12 zeros

Quadrillion has 15 zeros

Quintillion has 18 zeros

Sextillion has 21 zeros

Septillion has 24 zeros

Octillion has 27 zeros

Nonillion has 30 zeros

Decillion has 33 zeros

Undecillion has 36 zeros

Duodecillion has 39 zeros

Tredecillion has 42 zeros

Quattuordecillion has 45 zeros

Quindecillion has 48 zeros

Sexdecillion has 51 zeros

Septendecillion has 54 zeros

Octodecillion has 57 zeros

Novemdecillion has 60 zeros

Vigintillion has 63 zeros

Googol has 100 zeros.

Centillion has 303 zeros (except in Britain, where it has 600 zeros)

Googolplex has a googol of zeros

Differential Equations

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.

Differential Equations

What is Drakunov sliding mode observer?

It's a dynamical system that allows to reconstruct state vector of a nonlinear system using observation of the system output. See the Wikipedia article "State Observers".

Differential Equations
Root Beer

Where in SW Chicagoland in the 1970s was there an A and W root beer stand next to a Kentucky Fried Chicken stand?

AnswerI Believe it was on Kedzie....somewhere between 61st and 67th. I think an A@W restaurant is there now. AnswerOr was this on W.Columbus?.
Mathematical Finance
Numerical Analysis and Simulation
Differential Equations

How do you derive the main PDE in the Heston stochastic volatility model?

It is not possible to reproduce the equations on this website, however you can find a detailed derivation at the related link.

College Applications and Entrance Requirements
Math and Arithmetic
Differential Equations

What is convolution?

In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval. we mainly use impulse functions to convolute in dicreate cases

Math and Arithmetic
Differential Equations

What is the rules in multiplying polynomials?

If there is subtracting terms in either polynomial, change them to adding a negative. Each term in the first polynomial is multiplied by each term in the 2nd polynomial, then add all the resulting terms together (taking into account the signs of the resulting multiplications), simplify by combining like powers of the variable.

This is basically what you are doing when you multiply 2 numbers by hand, example: 997 x 42 = 41874


x 42

----- First you mutiply 2 x 7 = 14, put the 4 carry the 1, etc.

Imagine instead (4x + 2)(9x2 + 9x + 7) = 2*7 + 2*9*x + 2*9*x2 + (4*x)*7 + (4*x)*(9*x) + (4*x)*(9x2) = 14 + 46x + 54x2 + 36x3

For x = 10 --> 14 +460 + 5400 + 36000 = 41874.

Cocktail and Mixed Drink Recipes
Differential Equations

How many fifths of liquor to one gallon?


Math and Arithmetic
Differential Equations

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.
Mathematical Finance
Differential Equations

What does PDE mean?

In mathematics, a PDE is a Partial Differential Equation.

To partially differentiate an equation, read below:

Suppose you have a function f(x,y) and suppose you want to partially differentiate it w.r.t. x then you consider y as a constant and find d/dx(f(x,y)).

Eg. - Let f(x,y)=xy+x+y then on partially differentiating f(x,y) w.r.t. x -


= d/dx(xy) + d/dx(x) + d/dx(y)

= y(d/dx(x)) + 1 + 0 (as y is constant)

= y +1

Some application(s) of partial differential equations that I know -

1. Find the centre of a conic:

Suppose you have a curve as a function of x and y, say f(x,y).

Then to find its centre -

-> Partially differentiate f(x,y) w.r.t. x. Let the equation obtained be e1.

-> Partially differentiate f(x,y) w.r.t. y. Let the equation obtained be e2.

Solve e1 and e2 to get (x,y) which is the centre of the curve.

2. To find the (conservative) force acting on an object if its Potential energy is given as a function of distance:

-> Let the potential energy function be U(x,y).

-> To find the force acting on object in x-direction, find minus(partial derivative of U(x,y) w.r.t. x).

-> Same method to find force acting on the object in y-direction.

-> Only works for conservative force.

For more information, please see the related link.

English Language
Math and Arithmetic
Differential Equations

How many one-sixths make 2?


Math History
Differential Equations

What is the quotient of eighty-one and three?


Swimming Pools
Mathematical Analysis
Differential Equations

Will two 1.5-inch PVC pipes into a 2-inch PVC pipe flow as much water as 2-inch PVC into your pool pump and what would be the max gpm the pump could draw and would you need three 1.5-inch pipes?

I'm no pool expert but I can do the basic maths. I'd presume the limiting factor on how much water will pass through a pipe is its cross sectional area, and that these are circular pipes. If so, the area of a 1.5 inch diameter pipe is pi x .75 x .75, and of a 2inch diameter pipe is pi x 1 x 1 . So two of the smaller pipes will have a combined area of 1.125pi sq. inches, more than the single bigger pipe at 1pi sq. inches. Two 1.5" pipes into a single 2" line is acceptable for your flow rates. The flow rate will depend more on your pump than the two 1.5" diameter lines. You could add a third, or even more lines and individually isolate them with ball valves so you can adjust the flow from each as per the requirements of the pool.

Math and Arithmetic
Differential Equations

What does the word equation mean in math?

An equation means an expression with an equal sign in it (=), and that makes sense. For example 4 + 3 = 7 is an equation, while 4 + 3 = 6 or 4 + 5 are not equations. The first one is incorrect, while the second one is an expression.

The word equation describes a statement that two sets of something are equal to one another.

In mathematics, the two sets can consist of numbers, letters, or symbols.

In chemistry, an equation describes - in symbols - the changes that take place in a chemical reaction.

In everyday usage, an equation suggests one or more things are equal to one or more other things, or describe the varying aspects which need to be taken into consideration when analyzing a situation.

Home Equity and Refinancing
Math and Arithmetic
Differential Equations

Is it a wise choice to refinance a mortgage to save a hundred dollars a month?

Yes, that's usually the minimum savings amount that I recommend to someone who is contemplating to refinance. If you save $100 a month that equals $1,200 a year, which totals to $18,000 in 15 years and $36,000 in 30 years. Weigh that wersus the costs (usually no more than $6000, worse case scenario) and that to me seems like a wise decision. If you need any help with this feel free to call my office (214)607-1445.

American Cars
Weight and Mass
Mathematical Analysis
Differential Equations
How To

How to find the weight of 1sq.m of 5mm m.s plate?

steel weighs 490 pounds per cubic foot. A piece of steel 12" x 12" x 1" weighs 40.83 pounds. 7.65 lbs per square foot So ... 48" x 96" = 4' x 8' = 32 square feet and 7.65 lbs per square foot x 32 square feet = 245 lbs per full 0.1875" thick sheet

Quantum Mechanics
Waves Vibrations and Oscillations
Differential Equations

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 = -(2/2m)∂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:

(iψ)df/dt = -(2/2m)d2ψ/dx2f + Vψf

Dividing through by ψf gives:

(i)(1/f)df/dt = -(2/2m)(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/)

2) -(2/2m)d2ψ/dx2 + Vψ = , 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/) + C. Exponentiating and, since C will be absorbed later, removing C:

f(t) = e-(iEt/)

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/)

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.

Math and Arithmetic
Electronics Engineering
Differential Equations

What is a quadrillion?

Quadrillion may mean either of the two numbers (depends on long and short scales):

  • 1,000,000,000,000,000 (one thousand million million; 1015; SI prefix peta) - increasingly common meaning in English language usage.
  • 1,000,000,000,000,000,000,000,000 (1024; SI prefix yotta) - increasingly rare meaning in English language usage.

    It is 1,000,000,000,000,000.

  • Simply put, in the English language, it is "one thousand trillion."

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