Specific applications of differential equations, urgency, and applications of differential equations

The practical applications of differential equations are as follows:

First, start with a discrete sequence, define the limits of the sequence, whether it converges or diverges, the nature of the convergent sequence, the convergence criterion, and so on.

Equations with unknown quantities are equations, mathematics first developed in counting, and about numbers and unknowns are combined through addition, subtraction, multiplication, division and idempotency operations to form algebraic equations: unary equations, unary quadratic equations, binary linear equations, and so on. However, with the advent of the concept of functions.

As well as the introduction of function-based differentiation and integration operations, the scope of equations is broader, unknown quantities can be functions, vectors and other mathematical objects, and operations are no longer limited to addition, subtraction, multiplication and division.

Then the limit of the function is discussed, starting from the definition, the idea of the limit of the sequence is transferred, the nature of the limit of the function is discussed, etc., and the sequence and the function are connected by Heine's principle. After all, the number series is a discrete quantity (the number series can be understood as a function of a natural number), and the function is mainly a continuous quantity.

Since mathematics has changed from constant mathematics to variable mathematics, the content of equations has also been enriched because mathematics has introduced more concepts, more operations, and thus more equations. The development of other natural sciences, especially physics, has also directly put forward the need for equation solving, providing a large number of research topics.

Since the domain of a continuous function is a set of real numbers, and a sequence of numbers can be regarded as a function defined on a set of positive integers, the function introduces the continuous and uniform continuous gnator defined by the limit, and then gives the property theorem of bounded, zero or intermediate or maximum value of the continuous function.

A differential equation is an equation that contains an unknown function and its derivative. The unknown quantity of this type of equation is a function, and unlike the equation of the outer number of the outer number, there is a derivative operation for the unknown function, and it can be a higher-order derivative.

However, if the unknown function in an equation contains only one independent variable, then a differential equation is an ordinary differential equation.

c(1)=1 2 is a special solution of non-homogeneous order, and you can apply the formula to the homogeneous general solution. Find out that there are two arbitrary constants, and if the condition c(0)=0, list an equation [and then use c(0)=0 to get another equation] so that the two equations determine two constants.

The general solution formula of the homogeneous equation is also attached: The general solution formula is as follows: the system of homogeneous linear equations ax=0:

If x1, x2, xn-r are the basic solution system, then x=k1x1+k2x2+kn-rxn-r, which is the total solution of ax=0 (or the general solution of the system of equations).To find the general solution of the homogeneous dust-hu linear equations, we must first find the basic solution system: 1. Write the coefficient matrix a of the homogeneous equations; 2. Transform a into a ladder matrix through the primary row; 3. The variables corresponding to the non-principal element columns in the stepped matrix are taken as free elements (n r); d makes one of the free elements 1 and the rest 0, and we get n r solution vectors, which is a basic solution system.

It's not easy to code words, hope.

A differential equation is a relation that contains an unknown function and its derivative. Solving differential equations is to find out the unknown functions.

Differential equations were developed along with calculus. The founders of calculus, Newton and Leibniz, both dealt with problems related to differential equations in their writings. Differential equations have a wide range of applications and can solve many derivative-related problems.

Many kinematic and dynamic problems involving variable forces in physics, such as the falling motion of the air as a function of velocity, can be solved by differential equations. In addition, differential equations have applications in fields such as chemistry, engineering, economics, and demography.

This is not a differential equation, there is no differential operation. Rather, it is a system of difference equations.

Have you ever studied a system of linear difference equations?

This example is special because according to the second equation, ct can be represented by yt, and according to the first equation, it can also be represented by yt (replace ct with yt with the first equation).

Bring the above obtained it into the third equation to get the equation only about yt.

It looks like a system of equations, but it's just one equation...

The relationship between y(t+1) and yt is equivalent to the common notation a(n+1),an in a sequence.

In fact, only high school knowledge is sufficient.

According to Newton's heating and cooling theorem, t(t) satisfies dt dt = -k(t-20), where k>0, which denotes the proportionality constant, and "-denotes that the temperature of the object is decreasing. Solving the pure slip of the differential square row yields t(t)=ce -kt+20, and from t(0)=5, c=-15 is obtained. i.e. t(t) = 20-15e -kt

PS: The principle of heating and cooling of Newton's trousers, the rate of change of the temperature of the object is directly proportional to the difference between the temperature of the object and the ambient temperature.