Equations that relate independent variables, dependent variables, and their derivatives or differentials

Ordinary Differential Equations (ODE)

Partial Differential Equations

The order of an ordinary differential equation is determined by the order of the highest derivative (differential) in the equation.

The formula for y that makes the ordinary differential equation equal to zero

The solution of the equation contains arbitrary constants that are not related to each other, and the number is equal to the order of the equation

Taking a special value of any constant is generally derived from the initial value condition

An ordinary differential equation of order n contains an initial value condition

Find the value of a solution to an ordinary differential equation that satisfies the initial value condition

The first derivative has been solved

The right end is the multiplication of functions about x and y respectively.

1. Identify the type

2. Separate variables

3. Integrate both sides of the equal sign and add appropriate constants

4. Simplify and get the general solution

5. Determine whether there is an initial value condition

1. Order

2. Bring it into the equation to simplify and eliminate y

3. What is obtained is an ordinary differential equation of separable variables.

4. Solve according to the solution method of ordinary differential equations with separable variables

inhomogeneous term

Recorded as (1)

Recorded as (2)

First solve (2), and obtain its general solution by solving separable variable equations, which contains an arbitrary constant

constant change

Substitute the "general explanation" after the change into (1)

Solve first level General solutions to linear ordinary differential equations

constant variation method

You can also directly apply the formula in subsequent problem solving

α=0, is a first-order linear ordinary differential equation

α=1, is a separable variable type

1. Identify the type

2. Divide both sides by y raised to the α power at the same time

3. Let y raised to the (α-1) power equal to z

4. Transform into a first-order linear ordinary differential equation of z with respect to x

5. Solve according to the method of solving first-order linear ordinary differential equations

form

Reduction method

form

solution

form

solution

f(x) is a non-homogeneous term

L[cy]=cL[y]

L[y x]=L[y] L[x]

Assume that y1=y1(x), y2=y2(x) are both solutions to L[y]=0, c1 and c2 are constants, then c1y1 c2y2 is also a solution to L[y]=0

When , the two functions are linearly related, and c1 and c2 are related to each other.

When , the two functions are linearly independent, c1 and c2 are independent of each other, c1y1 c2y2 is the general solution of L[y]=0

Assume y1 and y2 are two linearly independent solutions of L[y]=0, c1 and c2 are arbitrary constants, then the general solution of L[y] is y=c1y1 c2y2

Suppose y`=y`(x) is a special solution of L[x]=f(x), and Y=c1y1 c2y2 is a general solution of L[y]=0. Then, the general solution of L[y]=f(x) is y` Y, c1, c1 are arbitrary constants

Assume that y1 and y2 are special solutions of L[x]=f1(x) and L[x]=f2(x) respectively, then y1 y2 is the solution of L[x]=f1(x) f2(x).