§ 2 Firstorder differential equations
1. Existence and Uniqueness of Solutions to FirstOrder Differential Equations
The general form of a firstorder 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 twovalued 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 firstorder 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 firstorder 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 . _{}_{}_{}