x= 1，y=0，z = 1； x= 1，y= 1，z = 0； x=0，y= 1，z= 1
∫a∈[0, 10] ∴ When a=0, we can get: x=-y=b(b is an arbitrary integer), z = 0；; When a= 1, we can get: x=-y=b(b is an arbitrary integer), z =1; When a=2, x=y= 1, z = 0；; When a=3, x = y = z =1; When a=4, there is no solution; When a=5, there is no solution; When a=6, x=2, y = z =-1; When a=7, x=2, y=0, z =-1; When a=8, x=2, y = z = 0; When a=9, x=2, y= 1, z = 0；; When a= 10, x=2, y=z= 1.
∴∑a=0+ 1+2+3+6+7+8+9+ 10=46
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Equations with unknowns are equations, and mathematics was first developed in counting. The combination of addition, subtraction, multiplication, division and idempotency between numbers and unknowns forms algebraic equations: one-dimensional linear equations, one-dimensional quadratic equations, two-dimensional linear equations and so on. However, with the emergence of the concept of function and the introduction of differential and integral operation based on function, the category of equation is wider, and the unknowns can be mathematical objects such as functions and vectors, and the operation is no longer limited to addition, subtraction, multiplication and division.
Equation occupies an important position in mathematics and seems to be an eternal topic in mathematics. The appearance of equations not only greatly expanded the application scope of mathematics, but also solved many problems that could not be solved by arithmetic problem-solving methods, which had a great influence on the progress of mathematics later. In particular, many important discoveries in mathematics are closely related to it.
The equations in middle schools are basically of this kind, and the unknowns in the equations can appear in the independent variables of fractions, algebras, roots, trigonometric functions and exponential functions.
We have the problem of solving equations in middle school. Generally speaking, we can convert equations into integral equations. Generally, it is transformed into a quadratic equation with one variable, or the solution of a system of linear equations with multiple variables.
Root formula of quadratic equation with one variable
Different from the above equation, the unknown in the equation is the function itself, not the independent variable of the function; Operations involve addition, subtraction, multiplication, division and function synthesis.
There is no unified theory and general method to solve the functional equation. For some functional equations, we can consider:
substitution method
Cauchy solution: a method of taking natural numbers, integer values, rational numbers and all real numbers from independent variables in turn to get the function value. Generally, the range of the solution will be limited when the function is continuous and monotonous.
Because mathematics has changed from constant mathematics to variable mathematics, the contents of equations are enriched, because mathematics has introduced more concepts and more operations, thus forming more equations. The development of other natural sciences, especially physics, also directly puts forward the demand of solving equations and provides a large number of research topics.
ordinary differential equation
Differential equation refers to an equation containing unknown functions and their derivatives. The unknowns of this kind of equations are functions. Different from functional equation, it has derivative operation on unknown function, which can be high-order derivative. But if the unknown function in the equation contains only one independent variable, then the differential equation is an ordinary differential equation.