Differential Equations

Over 9,000.

Multiply each monomial in the first polynomial with each monomial in the second polynomial. Then add everything up. This follows from the distributive property.

Thus, for example:

(a + b)(c + d)

= ac + ad + bc + bd

Often you can combine terms after adding:

(x + 3)(x + 5)

= x2 + 5x + 3x + 5

= x2 + 8x + 5

An ordinary differential equation is an equation relating the derivatives of a function to the function and the variable being differentiated against. For example, dy/dx=y+x would be an ordinary differential equation. This is as opposed to a partial differential equation which relates the partial derivatives of a function to the partial variables such as d²u/dx²=-d²u/dt².

In a linear ordinary differential equation, the various derivatives never get multiplied together, but they can get multiplied by the variable. For example, d²y/dx²+x*dy/dx=x would be a linear ordinary differential equation.

A nonlinear ordinary differential equation does not have this restriction and lets you chain as many derivatives together as you want. For example, d²y/dx² * dy/dx * y = x would be a perfectly valid example

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Liquid Volume Measuring Devices: The Graduated Cylinder and Buret Like weighing, measuring liquid volume is a fundamental and frequently encountered lab task. However, liquid volume is frequently measured using either a graduated cylinder or a buret. As the name implies, a graduated cylinder is a cylindrical glass (or plastic) tube sealed at one end with a calibrated scale etched (or marked) on the outside wall. Graduated cylinders come in a range of sizes (volume capacities), and much like a measuring cup, volume is measured by adding liquid to the cylinder and comparing the liquid level to the graduated scale. The measured volume corresponds to the volume of liquid contained in the cylinder. Hence, the graduated cylinder and devices like it (volumetric flasks, Erlenmeyer flasks, and beakers) are classified as to-contain (TC) devices. The volume of liquid in the graduated cylinder is obtained directly by reading the calibrated scale. In most situations, the liquid will be water or an aqueous solution.The liquid surface is curved (U-shaped) rather than horizontal due to the relatively strong attractive force between water and glass. (The curved surface is called the meniscus.) As a general rule, the bottom of the meniscus is taken as the liquid level in the cylinder (and any other volume measuring device). The scale divisions on a graduated cylinder are generally determined by its size. For example, the 50-mL graduated cylinder is divided into 1 mL increments. However, the scale of a 10-mL graduated cylinder is divided into 0.1 mL increments, and the scale of a 500-mL graduated cylinder is divided into 5 mL increments.The graduated cylinder scale is a ruled scale, and it is read like a ruler. The scale is read to one digit beyond the smallest scale division by estimating (interpolating) between these divisions. With a 50-mL graduated cylinder, read (and record) the volume to the nearest 0.1 mL. The 10-mL graduated cylinder scale is read to the nearest 0.01 mL and the 500-mL graduated cylinder scale is read to the nearest milliliter (1 mL).A buret is a scaled cylindrical tube attached to a stopcock (valve). A buret is designed to dispense or transfer a precisely measured volume of liquid to another container. The volume of liquid dispensed is determined by reading and recording the buret scale which corresponds to the liquid level in the buret before any liquid is transferred, Vinitial (or Vi),and after the transfer is complete, Vfinal (or Vf). The volume of liquid transferred is obtained by difference (Vf - Vi) and it is sometimes designated as Vt.Burets are available in a limited range of sizes; the most common size is 50-mL. The scale of a 50-mL buret is divided into 0.1 mL increments. Therefore, when the liquid level in a buret is read, it is read (and recorded) to the nearest 0.01 mL. Water or aqueous solutions are the most common liquids used with a buret, and like the graduated cylinder the bottom of the meniscus is taken as the liquid lever. The buret and devices like it (pipet and syringe) is classified as a to-deliver (TD) devices.

Partial differential equations are great in calculus for making multi-variable equations simpler to solve. Some problems do not have known derivatives or at least in certain levels in your studies, you don't possess the tools needed to find the derivative. So, using partial differential equations, you can break the problem up, and find the partial derivatives and integrals.

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Many real world problems can be represented by first order differential equation. Some applications of differential equation are radio-active decay and carbon dating, population growth and decay, warming/cooling law and draining a tank.

exact differential equation, is a type of differential equation that can be solved directly with out the use of any other special techniques in the subject. A first order differential equation is called exact differential equation ,if it is the result of a simple differentiation. A exact differential equation the general form P(x,y) y'+Q(x,y)=0
Differential equation is a mathematical equation. These equation have some fractions and variables with its derivatives.

I assume that you mean that you are given a differential equation dy/dx and want to solve it. If that is the case, then you would multiply by dx on both sides and then integrate both the left and right sides of the equation.

relation of cauchy riemann equation in other complex theorems

There are 6 triangles in an 8 sided octagon

It is a labor remuneration system where a worker's wage is increased proportionality with the number of physical units produce. This system differs from the straight rate labor system in that a worker is not paid a fixed flat rate for every unit of output but his earning per unit produced keeps on increasing as the worker produced more.

Normally there is a different rate for every level of output. Example

for the first 100 units -------------- wage; $10

for the next 101-200 units---------wage; $15

for the next 201-300 units--------- wage; $20

300 units and above----------------- wage; $25

This is a super sweating system that push the workers to their limited best ability as they pursue a higher wage rate.

Consistent equations are two or more equations that have the same solution.

Law of Detachment also known as Modus Ponens (MP) says that if p=>q is true and p is true, then q must be true.

The Law of Syllogism is also called the Law of Transitivity and states: if p=>q and q=>r are both true, then p=>r is true.

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(f(x,y))

= 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.

The website in the Related Link should be of some assistance.

It is an equation in which one of the terms is the instantaneous rate of change in one variable, with respect to another (ordinary differential equation). Higher order differential equations could contain rates of change in the rates of change (for example, acceleration is the rate of change in the rate of change of displacement with respect to time). There are also partial differential equations in which the rates of change are given in terms of two, or more, variables.

39 = 3 x 13

52 = 2^2 x 13

169 = 13^2

LCM is: 2^2 x 3 x 13^2 = 2028

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The order of a differential equation is a highest order of derivative in a differential equation.

For example, let us assume a differential expression like this.

d2y/dx2 + (dy/dx)3 + 8 = 0

In this differential equation, we are seeing highest derivative (d2y/dx2) and also seeing the highest power i.e 3 but it is power of lower derivative dy/dx.

According to the definition of differential equation, we should not consider highest power as order but should consider the highest derivative's power i.e 2 as order of the differential equation. Therefore, the order of the differential equation is second order.

differential centrifugation is a common procedure in microbiology and cytology used to separate certain organelles from whole cells for further analysis of specific parts of the cells.

Applications of ordinary differential equations are commonly used in the engineering field. The equation is used to find the relationship between the various parts of a bridge, as seen in the Euler-Bernoulli Beam Theory.