What is a linear equation and how to tell if an equation is linear or not

Linear equationsAlso called onceEquationsRefers to an equation in which all unknowns are one-time. Its general form is ax+by+.cz+d=0。

The essence of a linear equation is to multiply both sides of the equation by any identical non-zero number, and the essence of the equation is not affected.

Because in the Cartesian coordinate system.

The representation of any of the above primary equations is a straight line. Each term that makes up a primary equation must be a constant or the product of a constant and a variable. And the equation must contain a variable, because if there are no variables, the equation with only constants is algebraic, not an equation.

Linear equation form.

Addition, subtraction, and subtraction is the method of adding or subtracting two equations to eliminate one of the unknowns.

Usually, we multiply both sides of one equation by a number that is not 0 at the same time, so that one of the coefficients is the same as the corresponding coefficients of the other equation. Add or subtract the two equations.

The shape is ax+by+.cz+d=0, the linear equation for x and y, refers to the equation that can be deformed into ax+by+c=0 after finishing (where a, b, c are known numbers). The univariate linear equation is the simplest equation and is of the form ax=b.

Because the graph represented by a primary equation in a coordinate system is a straight line, it is called a linear equation.

The so-called linear differential equation lineardifferentialdifferentiation, in which a, can only appear the function itself, and the derivative of any order of the function; b. There can be no operation between the function itself and all derivatives except for addition and subtraction; c. The function itself and itself, and the derivative function of each order cannot have any operations other than addition and subtraction; d. It is not allowed to do any form of composite operation on the function itself and each derivative, such as: siny, cosy, tany, root y, lny, lgx, y, y, y x, x y. If the above conditions cannot be compounded, it is a nonlinear equation

For example: y'=sin(x)y is linear but y'=y 2 is not linear Note two things: (1) y'The former coefficient cannot contain y, but it can contain x, such as:

y*y'=2 is not linear x*y'=2 is linear (2) The coefficient before y cannot contain y, but it can contain x, such as: y'=sin(x)y is linear y'=sin(y)y is nonlinear.

There are roughly three conditions:

The unknown function and its derivatives are all powers of the first order.

The coefficients of unknown functions and derivatives can only contain independent variables or constants, which are also covered in the first order of linear differential equations. dy dx p(x)y ten q(x), where p(x) is the coefficient of the unknown function with independent variables.

There can be no composite function forms of unknown functions and derivatives of each order. For example, sinxdx cosydy, cosy, which is a composite function, is not a linear differential equation.

Differential equations are mathematical equations used to describe the relationship between a certain class of functions and their derivatives, and in algebraic equations in elementary mathematics, the solution is a constant value.

Differential equations can be divided into ordinary differential equations and partial differential equations. It has a wide range of applications in fields such as chemistry, engineering, economics, and demography.

Linear and non-linear:

Both ordinary differential equations and partial differential equations can be divided into linear and nonlinear equations.

If there are no independent variables and squares or other product terms of the differential terms, nor the product of the strain and their differential terms, the differential equation is a linear differential equation, otherwise it is a nonlinear differential equation.

A homogeneous linear differential equation is a finer classification of a linear differential equation in which the solution of the differential equation is multiplied by the previous coefficient or added to another solution is still the solution of the differential equation.

If the coefficients of a linear differential equation are constant, it is a linear differential equation with constant coefficients. Linear differential equations with constant coefficients can be converted into algebraic equations using the Rasley transformation, thus simplifying the process of solving.

For nonlinear differential equations, there are only a few methods to obtain analytical solutions to differential equations, and these methods require special symmetry in differential equations. Nonlinear differential equations can be very complex over a long period of time, or they can be chaotic.

For first-order differential equations, the shape is: y'+p(x)y+q(x)=0 is called"Linear"。

For second-order differential equations, the shape is: y''+p(x)y'+q(x)y+f(x)=0"Linear"。

For example: y'=sin(x)y is linear but y'=y 2 is not linear.

Note two points: 1) y'The previous coefficient cannot contain y, but it can contain x, such as: y*y'=2 is not linear; x*y'=2 is linear.

2) The coefficient before y cannot contain y, but it can contain x, such as: y'=sin(x)y is linear y'=sin(y)y is nonlinear.

3) In the whole equation, only y and y can appear', sin(y), y 2, y 3, etc., such as: y'=y is linear; y'=y 2 is nonlinear.

The form is ax+by+.cz+d=0。The essence of a linear equation is to multiply both sides of the equation by any identical non-zero number, and the essence of the equation is not affected by the precedent of the equation.

A differential equation is called a linear differential equation if it contains only an unknown function and its derivatives as a power of the whole. Otherwise, it is called a nonlinear differential equation.

It can be understood that the unknown Buhui function y in this differential equation is not more than once, and the derivatives of y in this equation should also be no more than once. In algebraic equations, equations with only unknown numbers are called linear equations.

The function of this equation is represented as a straight line, so it is called a linear equation. It can be understood as: that is, the highest order term of the equation is -order, and 0 terms are allowed, but not more than once. For example, ax+by+c=0, where c is the 0th term for x or y.

Take second-order differential equations as an example (and so on for higher-order ones): after simplification, they can be deformed into this form called linear differential equations: p(x)y"+q(x)y'+r(x)y=s(x)

where p(x), q(x), r(x), and s(x) are all functions of known x).

No matter how simplified, a non-primary square differential equation with a y or y derivative in the equation is a nonlinear differential equation.

For example: y'y=y, although y is not a one-time square, but I can become y(y) by equivalence deformation'-y)=0, i.e. y=0 or y'-y=0 because y and y'are all one-time squares, so they are linear differential equations. And their coefficients are all constants, so they can be called constant coefficient differential equations.

Another example is (sinx)y'-y=0 because y'and y are all 1 degrees (the function term containing x is not counted), so it is a linear differential equation. And y'The coefficient of is sinx and is therefore a linear ordinary differential equation with variable coefficients.

Another example is y'y=1, no matter how simplified (e.g. by y) it cannot become y'and y degrees are both of the form 1, so the equation is a nonlinear differential equation.

One more sentence: Linear differential equations have analytic solutions, that is, they can be written in the form of the analytic expression y=f(x). But nonlinear differential equations are difficult to say.

In general, some first-order nonlinear differential equations have analytical solutions. However, it is difficult to have an analytical solution to a nonlinear differential equation of the second order or above.

Linear equations: Algebraic equations, such as y=2x+7, where any variable is a power of one. The graph of this equation is a straight line, so it is called a linear equation.

The so-called nonlinear equation means that the relationship between the dependent variable and the independent variable is not linear, and there are many such equations, such as square relationship, logarithmic relationship, exponential relationship, trigonometric relationship, and so on. It is often difficult to get an exact solution to such equations, and it is often necessary to find an approximate solution to the problem. The corresponding method of finding the approximate solution has gradually received everyone's attention.

The basic method for finding the approximate solution of a nonlinear equation is the iterative method, which is a method that gradually approaches the exact solution.

You're nonlinear.

In an ordinary differential equation, if the right-hand function f is once for the whole of the unknown function y and its mediated derivatives y',y'',y(n)(n-mediated derivatives), it is a linear ordinary differential equation, otherwise it is called a nonlinear ordinary differential equation. y’‘+yy'=x is nonlinear. y’+y+y''=x is the current one.

To learn ordinary differential equations well, you must first listen carefully to the lectures and grasp the basic definitions. The solution method of differential equations is very important, and the various types of equations should be resolved back, and the corresponding solutions should be memorized. To solve equations, as long as you master the formulas, you can basically solve the exam questions.

Of course, you also need to do certain questions and be proficient in various calculation skills.