The first order ordinary differential equation is a common differential equation, and its form is y'=f(x, y). This kind of equation is widely used in nature, engineering, social science and other fields. There are many methods to solve first-order ordinary differential equations, including separation of variables, integral factor method, method of substitution method, constant variation method and so on.

These methods will be described in detail below.

Variable separation method

The method of separating variables is to express the unknown function and its derivative in brackets, and then transform the equation into two independent differential equations according to the relationship between function and derivative, so as to solve it.

For example, for the equation dy/dx=xy, it can be transformed into d(y/x)=y/x*dx, and then the two sides are integrated to get Y/X = X 2/2+C, where C is a constant. By separating variables, we can get the general solution of this equation.

Integral factor method

The integral factor method is to find an equation with the same solution as the equation y'=f(x, y) and convert it into an integral x on both sides of an equation, so as to solve it.

For example, for the equation Dy/DX = E (X-Y), we can find an equation Dy/DX+Y = E X with the same solution. Then let z=y+C (where c is a constant), then the original equation is transformed into dz/dx = e x, and finally the two sides are integrated to get z = e x+C 1 (where C 1 is a constant), that is, y = e (-x)+c 1. Through the integral factor method, we can get the general solution of this equation.

substitution method

The substitution method is to express the derivative of the original equation in brackets, and then replace the expression in brackets with a function, thus transforming it into an equation.

For example, the equation dy/dx = (x+y) 2, and the derivative can be expressed in brackets to get (d/dx) (y-x 2) = 0. Then let z = y-x 2, and the original equation is transformed into z'=0. Finally, by substitution, we can get the general solution of this equation.

Constant variation method

Constant variational method is to replace the unknown function and its derivative of the original equation with a constant, and then transform it into an equation to solve it. For example, for the equation dy/dx=xy, we can make Z = Y-AX 2 (where a is a constant), and the original equation can be transformed into Z'/A = X 2. Then integrate the two sides to get z = a * x 3/3+c (where c is a constant). By the method of constant variation, we can get the general solution of this equation.

In addition to the above four methods, power series method and numerical calculation method can also be used to solve first-order ordinary differential equations. No matter which method is adopted, it is necessary to have a certain understanding of differential equations and master mathematical methods to solve first-order ordinary differential equations. At the same time, it is also necessary to choose appropriate methods to solve specific problems according to specific conditions.