§ 2 First-order differential equations

1. Existence and Uniqueness of Solutions to First-Order Differential Equations

The general form of a first-order differential equation is

If on the region under consideration , then according to the existence theorem of implicit functions (Chapter V § 3 , 4, 2 ), the solution yields

or written in symmetrical form

[ Theorem of Existence and Uniqueness of Solutions ]   Given a Differential Equation

and initial value .

Set in closed area :

is continuous, then there is at least one solution to the equation, which takes value at and is deterministic and continuous in a certain interval included (this theorem is called Cauchy's existence theorem) .

If the inner pair variable also satisfies the Lipschitz condition, that is, there is a positive number , such that for any two-valued sum of the inner pair , the following inequality holds:

Then this solution is unique .

Two, integrable types and their general solutions

( C is an arbitrary constant in the table)

 Equation Type Solution points and general solution expressions 1 . Variable Separation Equations   f 1 ( x ) g 1 ( y )d x + f 2 ( x ) g 2 ( y )d y =0 Separate the variables, divide both sides by g 1 ( y ) f 2 ( x ) , and integrate separately . 2. Homogeneous equations        General assumptions then the variable is separable and belongs to type 1 make Substitute into the original equation to get the equation of the new unknown function u about the independent variable x :               x d u = [ F ( u ) – u ]d x Then solve for type 1 . 3 . Linear equation   Equation Type First find the corresponding homogeneous linear equation          Solution points and general solution expressions When q ( x ) ≡ 0 , it is called a homogeneous linear equation, and when , it is called an inhomogeneous linear equation general solution    Reusing the method of constant variation ( § 3, 2 , 2 of this chapter ), let Calculate and substitute into the original inhomogeneous linear equation , we can get 4 . Bernoulli equation Use variable substitution to transform the original equation into a linear equation about the new unknown function , and then solve it according to type 3 . 5 . Full (proper) differential equations   M ( x , y )d x + N ( x , y )d y =0 where M and N satisfy The equation can be written as     M ( x , y )d x + N ( x,y )d y =d U ( x,y )=0 where d U is the total (proper) differential . 6 . Equations that can be solved for y        y = F ( x,p ) in the formula Taking the derivative of both sides of the equation with respect to x , we get or             If the general solution of this equation can be found or , then the original equation can be solved . [ Lagrange equations ]       y = xf 1 ( p ) + f 2 ( p ) where is a known differentiable function [ Clero Equation ]       y = xp + F ( p ) where is a known differentiable function   Equation Type A linear equation that can be reduced to x Then solve according to type 3   turn into an equation Let , that is, p = c , and substitute it into the original equation . Solution points and general solution expressions ( see § 2, 3 ) 7. Equations that can be solved for x       x = F(y, p) in the formula Taking the derivative of both sides of the equation with respect to x , use If the general solution of this equation can be found    Then the original equation can be solved . 8. Equations without explicit unknown functions By introducing the appropriate parameter t , the original equation is transformed into 9. Equations without explicit independent variables Introducing the parameter t , the original equation is 10 . Equations that can be reduced to separable or homogeneous equations       Equation Type ( a ) Let z = ax + by + c , convert the original equation to type 1 ( b ) If the determinant Introduce new variables where α and β satisfy the equations Solution points and general solution expressions Then the original equation is transformed into a homogeneous equation ( type 2): If =0, b 1 ≠ 0 , then let z = a 1 x + b 1 y + c 1 ; If =0, b 2 ≠ 0, then let z = a 2 x + b 2 y + c 2, So the original equation is reduced to type 1. 11. The Riccati equation If it is known that the original equation has a particular solution y=y 1 ( x ) , make the transformation The original equation can be transformed into a linear equation ( type 3) : Or use the transformation y = y 1 ( x ) + u to convert to Bernoulli's equation ( type 4): Then solve according to type 3 and type 4 respectively . 12. Equations with integral factors M ( x, y ) d x + N ( x, y ) d y = 0 in the formula But there exists μ ( x, y ) that satisfies μ ( x, y ) is called the integral factor of the original equation Find the integral factor μ ( x, y ), and then solve it according to type 5. The method of finding the integral factor is shown in the table below .

How to find the integral factor

 condition Integration factor μ ( x, y ) condition Integration factor μ ( x, y ) xM+yN =0     xM+yN ≠ 0 condition Integration factor μ ( x, y ) condition of the form m ( x ) n ( y )         Integration factor μ ( x, y ) M,N are homogeneous forms of the same degree   M(x, y) = yM 1 (xy) N(x, y) = xN 1 (xy) there is suitable The constants m and n of ( determined by the method of comparison coefficients )   That is, M+iN is an analytic function of x+iy in the simply connected region that satisfies the differential equation x m y n

3. Strange solutions and their solutions

[ Singular solution of differential equation ]   The envelope of a family of integral curves (general solutions) of a differential equation is called the singular solution of this differential equation . A singular solution is the solution of the equation, and there is more than one integral curve passing through each point on the singular solution. , that is, at every point on the singular solution, the solution of the equation is not unique .

[ c - discriminant curve method ]   Let the general solution of the first-order differential equation be , where c is an arbitrary constant, and c is regarded as a parameter . From the following equations

All the curves obtained by eliminating c are called the c - discriminant curve of the curve family, which contains the envelope of the curve family . However, it should be noted that the c - discriminant curve is not necessarily the envelope of the curve family. Check .

Example to find a first order differential equation

general and singular solutions .

Solve the equation as

Let y '= p . Taking the derivative of both sides of the equation with respect to p , we get

So there is

which is

Substitute into the original equation and get a general solution

from

Eliminate c from , and get the c- discriminant curve y=x sum . Substitute it into the original equation directly, we know that y=x is not the solution of the known equation, so it is not a singular solution, but odd solution .

[ p - discriminant curve method ] For a first-order differential equation  , let , then the singular solution of the equation must be included in the following equations

In the curve obtained after eliminating p (called p - discriminant curve) . As for whether the p - discriminant curve is a singular solution, it also needs to be actually tested .

Example of a Differential Equation

strange solution .

Resolve _

Eliminate p to get the p- discriminant curve , that is, y= . Substitute it into the original equation to know that y= is a singular solution .