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MindMap Gallery differential equations

differential equations

Summary of knowledge points related to differential equations in advanced mathematics, including basic concepts of differential equations, differential equations of separable variables, variable substitution method, first-order linear differential equations, etc.

Edited at 2021-09-16 10:34:04

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differential equations
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differential equations

basic concept

Definitions and related concepts of differential equations

Definition: Any equation that represents an unknown function, the relationship between the derivative of the unknown function and the independent variables is called an ordinary differential equation.

Note: The derivative of the unknown function must appear

Definition: The order of the highest derivative of the unknown function appearing in a differential equation is called the order of the differential equation.

Definition: Substituting a function that makes a differential equation an identity is called the solution of a differential equation

Two different forms of solution

The solution contains any constants and the number of constants is the same as the order. Such a solution is called a general solution or a general solution.

The solution obtained by determining any constant from the general solution according to the specific conditions given by the problem is called a special solution

Differential Equations with Separable Variables

variable substitution method

homogeneous equation

can be reduced to a homogeneous equation

solution:

First order linear differential equation

linear equation

definition·

Call a first-order non-homogeneous linear equation

We call the first-order homogeneous linear equation relative to (1)

solution:

constant variation method

Method of converting constants in the general solution of homogeneous equations into undetermined functions

Bernoulli equation

When n=0, it is a linear equation

When n=1, it is a separable equation.

Solution: Although equation (10) is not a linear equation, it can be converted into a linear equation through variable substitution.

total differential equations

definition

Measure to judge

solution:

Applying curve integrals is path independent

Use indefinite integral to find u(x,y)

A method of directly summing up the differentials

integration factor

definition:

Note: The integration factor is generally not easy to obtain, but it can be observed in simple situations.

Remember to sum up the differential

Reducible higher-order differential equations

Definition: Differential equations of second order and above are called higher-order differential equations

Solution: successive integration

Structure of higher-order linear differential equations and their solutions

Structure of n-order linear differential equations and their solutions

is the coefficient function, f(x) is the free term

When f(x)¹0, it is called n-order non-homogeneous linear equation

When f(x)=0, it is called a homogeneous linear equation of order n

Characteristics of linear equations

All once

Both the coefficient function and the free term are functions of X

The concepts of linear dependence and independence of function groups

definition

A set of functions (A) defined in I: y1(x), y2(x),...,yn(x), if there are constants k1, k2,..., kn that are not all 0, such that k1y1 k2y2 ... knyn=0 xÎI, then (A) is said to be linearly dependent within I, otherwise it is said to be linearly independent

Notice

The case of two functions

y1(x), y2(x) are irrelevant

Structure of solutions to second-order linear differential equations

Structure of solutions to second-order homogeneous equations

Theorem 1

If the functions y1(x) and y2(x) are two solutions to equation (1), then y=C1y1 C2y2 is also a solution to (1). (C1, C2 are constants)

Theorem 2

If y1(x) and y2(x) are two linearly independent special solutions of equation (1), then y=C1y1 C2y2 is the general solution of equation (1)

Note

Theorem 2 can be extended to the case of n-order equations

example

Structure of solutions to second-order non-homogeneous linear equations

Theorem 3

Suppose y* is a special solution of the second-order non-homogeneous linear equation, Y is the general solution of the homogeneous equation (1) corresponding to (2), then y=Y y* is the second-order non-homogeneous linear differential equation (2 ) general explanation

example

According to Theorem 3, the steps for finding the general solution to the non-homogeneous equation (2) are:

Find the general solution Y of the homogeneous equation corresponding to (1)

Find a special solution y* of (2)

y=Y y*

Theorem 4

Assume that the right-hand side f(x) of the non-homogeneous equation (2) is the sum of several functions, such as

The special solution of , then y1* y2* is the special solution of the original equation

superposition principle of solutions

price reduction method and constant variation method

Finding linearly independent special solutions to homogeneous linear equations

Reduction method

Suppose y1 is a non-zero special solution of equation (1), let y2=u(x)y1 be brought into (1), we get

Liouville formula

The general solution of the homogeneous equation is

example

constant variation method

Solution to Homogeneous Linear Equations of Second Order with Constant Coefficients

definition

Standard form of linear differential equations with constant coefficients of order n

Standard form of second-order homogeneous linear equations with constant coefficients

Standard form of second-order non-homogeneous linear equations with constant coefficients

Solution of Homogeneous Linear Equations of Second Order with Constant Coefficients

solution

Find two linearly independent

observe

Therefore, r^2 pr q=0 (2)

characteristic equation

Roots of the characteristic equation

characteristic root

There are two unequal real roots (D>0)

Two linearly independent special solutions

The general solution of the homogeneous equation is

There are two equal real roots (D=0)

The characteristic root is r1=r2=-p/2

A special interpretation is

Another special solution is set to

The general solution to the homogeneous equation is

There is a pair of conjugate complex roots (D<0)

The characteristic root is

Regroup

The general solution to the homogeneous equation is

Note (2) In the formula, r^2, the coefficients and constant terms of r are y", y', and the coefficient of y in order

definition

The method of determining the general solution of a homogeneous linear equation with constant coefficients from the roots of its characteristic equation is called the characteristic equation method.

Write the characteristic equation of (1): r^2 pr q=0--(2)

Find the characteristic roots r1, r2 of (2)

According to the characteristic root situation, the general solution can be divided into three situations:

Solution of n-order homogeneous linear equations with constant coefficients

The characteristic equation is

Notice

Solution to non-homogeneous linear equations with constant coefficients of second order

Two forms of free term f(x)

(l is a constant)

polynomial of degree m

special shape

l=0ðf(x)=Pm(x)

Pm(x)=1ðf(x)=e^lx

l-real number aðf(x)=Pm(x)e^ax

l=a ibðf(x)=Pm(x)e^(a ib)

(a, b are real constants)

(One of them can be 0)

special shape

a=0

method

Find y* using undetermined coefficient method

Find a special solution Y* of y" py' qy=e^lxPm(x)

Let y*=Q(x)e^lx--(A)

Q(x) is a polynomial of x

If l is not a characteristic root of (2): l^2 pl q¹0

If l is the characteristic single root of (2): l^2 pl q=0

If l is a multiple root of (2): l^2 pl q=0, 2l p=0

step

Find the characteristic roots of the homogeneous equation corresponding to (1)

set up

y*=y1* y2*

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