n-th order
Theorem. If y 0- solution of a homogeneous equation L[y]=0, y 1- solution of the corresponding inhomogeneous equation L[y] = f(x), then the sum y 0 +y 1 is the solution to this inhomogeneous equation.
The structure of the general solution of the inhomogeneous equation is determined by the following theorem.
Theorem. If Y- particular solution of the equation L[y] = f(x) with continuous coefficients, 
- general solution of the corresponding homogeneous equation L[y] = 0, then the general solution of this inhomogeneous equation is determined by the formula
Comment. To write down the general solution of a linear inhomogeneous equation, it is necessary to find some particular solution to this equation and a general solution to the corresponding homogeneous equation.
Linear inhomogeneous equations n
Consider the linear inhomogeneous equation n-th order with constant coefficients
Where a 1, a 2, …, a n- real numbers. Let us write the corresponding homogeneous equation
The general solution of the inhomogeneous equation is determined by the formula
General solution of a homogeneous equation y 0 we can find, a particular solution Y can be found by the method of indefinite coefficients in the following simple cases:
In the general case, the method of varying arbitrary constants is used.
Method of variation of arbitrary constants
Consider the linear inhomogeneous equation n-th order with variable coefficients
If finding a particular solution to this equation turns out to be difficult, but the general solution to the corresponding homogeneous equation is known, then the general solution to the inhomogeneous equation can be found method of variation of arbitrary constants.
Let the corresponding homogeneous equation
has a general solution
We will look for a general solution to the inhomogeneous equation in the form
Where y 1 =y 1 (x), y 2 =y 2 (x), …, y n =y n (x) are linearly independent solutions of a homogeneous equation included in its general solution, and C 1 (x), C2(x), …, Cn(x)- unknown functions. To find these functions, let's subject them to some conditions.
Let's find the derivative
We require that the sum in the second bracket equals zero, that is
Let's find the second derivative
and we will demand that
Continuing a similar process, we get
In this case, one cannot require that the sum in the second bracket vanish, since the functions C 1 (x), C2(x), …, Cn(x) already subordinated n-1 conditions, but you still need to satisfy the original inhomogeneous equation.
Equations solved by direct integration
Consider the following differential equation:
.
We integrate n times.
;
;
and so on. You can also use the formula:
.
See Differential equations that can be solved directly integration > > >
Equations that do not explicitly contain the dependent variable y
The substitution lowers the order of the equation by one. Here is a function from .
See Differential equations of higher orders that do not contain a function explicitly > > >
Equations that do not explicitly include the independent variable x
.
We consider that is a function of . Then
.
Similarly for other derivatives. As a result, the order of the equation is reduced by one.
See Differential equations of higher orders that do not contain an explicit variable > > >
Equations homogeneous with respect to y, y′, y′′, ...
To solve this equation, we make the substitution
,
where is a function of . Then
.
We similarly transform derivatives, etc. As a result, the order of the equation is reduced by one.
See Higher-order differential equations that are homogeneous with respect to a function and its derivatives > > >
Linear differential equations of higher orders
Let's consider linear homogeneous differential equation of nth order:
(1)
,
where are functions of the independent variable. Let there be n linearly independent solutions to this equation. Then the general solution to equation (1) has the form:
(2)
,
where are arbitrary constants. The functions themselves form a fundamental system of solutions.
Fundamental solution system of a linear homogeneous equation of the nth order are n linearly independent solutions to this equation.
Let's consider linear inhomogeneous differential equation of nth order:
.
Let there be a particular (any) solution to this equation. Then the general solution has the form:
,
where is the general solution of the homogeneous equation (1).
Linear differential equations with constant coefficients and reducible to them
Linear homogeneous equations with constant coefficients
These are equations of the form:
(3)
.
Here are real numbers. To find a general solution to this equation, we need to find n linearly independent solutions that form a fundamental system of solutions. Then the general solution is determined by formula (2):
(2)
.
We are looking for a solution in the form . We get characteristic equation:
(4)
.
If this equation has various roots, then the fundamental system of solutions has the form:
.
If available complex root
,
then there is also a complex conjugate root. These two roots correspond to solutions and , which we include in the fundamental system instead of complex solutions and .
Multiples of roots multiplicities correspond to linearly independent solutions: .
Multiples of complex roots multiplicities and their complex conjugate values correspond to linearly independent solutions:
.
Linear inhomogeneous equations with a special inhomogeneous part
Consider an equation of the form
,
where are polynomials of degrees s 1
and s 2
; - permanent.
First we look for a general solution to the homogeneous equation (3). If the characteristic equation (4) does not contain root, then we look for a particular solution in the form:
,
Where
;
;
s - greatest of s 1
and s 2
.
If the characteristic equation (4) has a root multiplicity, then we look for a particular solution in the form:
.
After this we get the general solution:
.
Linear inhomogeneous equations with constant coefficients
There are three possible solutions here.
1)
Bernoulli method.
First, we find any nonzero solution to the homogeneous equation
.
Then we make the substitution
,
where is a function of the variable x. We obtain a differential equation for u, which contains only derivatives of u with respect to x. Carrying out the substitution, we obtain the equation n - 1
- th order.
2)
Linear substitution method.
Let's make a substitution
,
where is one of the roots of the characteristic equation (4). As a result, we obtain a linear inhomogeneous equation with constant coefficients of order . Consistently applying this substitution, we reduce the original equation to a first-order equation.
3)
Method of variation of Lagrange constants.
In this method, we first solve the homogeneous equation (3). His solution looks like:
(2)
.
We further assume that the constants are functions of the variable x. Then the solution to the original equation has the form:
,
where are unknown functions. Substituting into the original equation and imposing some restrictions, we obtain equations from which we can find the type of functions.
Euler's equation
It reduces to a linear equation with constant coefficients by substitution:
.
However, to solve the Euler equation, there is no need to make such a substitution. You can immediately look for a solution to the homogeneous equation in the form
.
As a result, we obtain the same rules as for an equation with constant coefficients, in which instead of a variable you need to substitute .
References:
V.V. Stepanov, Course of differential equations, "LKI", 2015.
N.M. Gunther, R.O. Kuzmin, Collection of problems in higher mathematics, “Lan”, 2003.
Differential equationsn-th order.
If the equation is solvable with respect to the highest derivative, then it has the form (1). An nth order equation can also be represented as a system of n first order equations.
(3)
For an n-th order equation, the conditions of the theorem on existence and uniqueness for the system are satisfied since (1)~(2)~(3).
The simplest cases of order reduction.
The equation does not contain the required function and its derivative up to the order k -1 inclusive , that is
In this case the order can be reduced to 
replacement. If we express this equation then the solution y can be determined by the k-fold integrable function p.
Example. 
.
Equation containing no unknown variable
(5)
In this case, the order can be lowered by one by substitution.
Example. 
.
Left side of the equation
(6)
is the derivative of some differential expression (
n
-1)th order
.
. If 
- a solution to the last equation therefore exists. We obtained the first integral of equation (6) and lowered the degree of the equation being solved by one.
Comment. Sometimes the left side of (6) becomes the derivative of a (n-1)th order differential equation only when multiplied by 
therefore, unnecessary solutions may appear here (reversing 
to zero) or we may lose the solution if 
discontinuous function.
Example. 
The equation
(7)
homogeneous relatively 
and its derivatives
.
Or where is the indicator 
is determined from the conditions of homogeneity.
The order of this equation can be lowered by one by replacing: .
If we substitute these relations into (7) and take into account the homogeneity of the function F , then in the end we get: .
Example. 
.
Second order differential equations,
allowing for a reduction in order.
Substitution 
.
If equation (8) can be resolved with respect to the highest derivative, then Eq. 
integrated twice over the variable x.
You can introduce a parameter and replace equation (8) with its parametric representation: 
. Using the relation for differentials: 
, we get: and
II
.
(9)
Let's use the parametric representation:
III.
.
(10)
You can lower the order by replacing: 
.
If equation (10) is solvable with respect to the highest derivative 
, then multiply the right and left sides by 
. We get: This is an equation with separable variables: 
.
Equation (10) can be replaced by its parametric representation: . Let's use the properties of the differential:.
Example. 
.
Linear differential equationsn-th order.
Definition.
Linear differential equations
n
-th order
are called equations of the form: 
. (1)
If the odds 
continuous for 
, then in the neighborhood of any initial values of the form:, where 
belongs to the interval, then in the neighborhood of these initial values the conditions are satisfied existence and uniqueness theorems. The linearity and homogeneity of equation (1) is preserved under any transformation 
, Where 
is an arbitrary ntimes differentiable function. Moreover 
. Linearity and homogeneity are preserved when the unknown function is transformed linearly and homogeneously.
Let us introduce a linear differential operator: , then (1) can be written as follows: 
. Wronski's determinant for 
will look like:
, Where 
- linearly independent solutions to equation (1).
Theorem 1. If linearly independent functions 
is a solution to a linear homogeneous equation (1) with continuous 
coefficients 
, then the Wronski determinant 
does not vanish at any point on the segment 
.
Theorem 2. The general solution of the linear homogeneous equation (1) with continuous 
coefficients 
there will be a linear combination of solutions 
, that is 
(2), where 
linearly independent on the segment 
private solutions (1).
(proved similarly to the case of a system of linear differential equations)
Consequence. The maximum number of linearly independent solutions to (1) is equal to its order.
Knowing one non-trivial particular solution to equation (1) - 
, you can make a substitution 
and lower the order of the equation while maintaining its linearity and heterogeneity. Usually this substitution is split into two. Since this is a linearly homogeneous representation, it preserves the linearity and homogeneity of (1), which means (1) must be reduced to the form. The decision 
by virtue of 
corresponds to the solution 
, and therefore 
. Having made a replacement 
, we obtain an equation with the order 
.
Lemma. (3)
Two equations of the form (3) and (4), where Q i and P i are continuous functions that have a common fundamental system of solutions, coincide, i.e. Q i (x)= P i (x), i=1,2,…n, x
Based on the lemma, we can conclude that the fundamental system of solutions y 1 y 2 …y n completely determines the linear homogeneous equation (3).
Let us find the form of equation (3), which has a fundamental system of solutions y 1 y 2 …y n . Any solution y(x) equation (3) linearly depends on the fundamental system of solutions, which means that W=0. Let us expand the Wronski determinant W over the last column.
Equation (5) is the desired linear differential equation having a given system of fundamental solutions. We can divide (5) by W, because it is not equal to zero x. Then:
(*)
According to the rule of differentiation of the determinant, the derivative of the determinant is equal to the sum of i=1,2...n determinants, the i-th row of each of which is equal to the derivative of the i-th row of the original determinant. In this sum, all determinants except the last one are equal to zero (since they have two identical lines), and the last one is equal to (*). Thus, we get:
, Then: 
(6)
(7)
Definition. Formulas (6) and (7) are called Ostrogradsky-Liouville formulas.
We use (7) to integrate a second-order linear homogeneous equation. And let us know one of the solutions y 1 of equation (8).
According to (7), any solution (8) must satisfy the following relation:
(9)
Let's use the integrating factor method.
Linear homogeneous equations with
constant coefficients.
If in a linear homogeneous equation all coefficients are constant,
a 0 y (n) +a 1 y (n-1) +….+a n y=0, (1)
then particular solutions (1) can be defined as: y=e kx, where k is a constant.
a 0 k n e kx +a 1 k n-1 e kx +….+a n k 0 e kx =0 a 0 k n +a 1 k n-1 +….+a n =0 (3)
Definition. (3) - characteristic equation.
The type of solution (1) is determined by the roots of the characteristic equation (3).
1). All roots are real and distinct , Then:
2). If all coefficients are real, then the roots can be complex conjugate .
k 1 =+i k 2 =-i
Then the solutions have the form:
According to the theorem: if an operator with real coefficients has complex conjugate solutions, then their real and imaginary parts are also solutions. Then:
Example.
Let us present the solution in the form 
, then the characteristic equation has the form:
, we get two solutions:
then the required function is:
3). There are multiple roots:
k
i
with multiplicity
i
.
In this case, the number of different solutions 
will be smaller, therefore, you need to look for the missing linearly independent solutions in a different form. For example:
Proof:
Let's say k i =0, if we substitute it into (3), we get that , then:
- particular solutions (3).
Let k i 0, let’s make the replacement 
(6)
Substituting (6) into (1), we obtain with respect to z a linear homogeneous equation of the nth order with constant coefficients (7).
The roots (3) differ from the roots of the characteristic equation (7) by the term k i .
(8)
If k=k i , then this k corresponds to the solution of equation (7) with root p=0, i.e. correspond to solutions of the form z= 
, then y= is the solution to equation (1). And the general solution looks like:
solution for k i
Euler's equation.
Definition. Equation of the form:
a i are constant coefficients, called Euler's equation.
Euler's equation by replacing x=e t is reduced to a linear homogeneous equation with constant coefficients.
You can look for solutions in the form y=x k, then they have the form:
Linear inhomogeneous equations.
If a 0 (x)0, then dividing equation (1) by this coefficient, we obtain:
.
If i and f are continuous on b, then (2) has a unique solution that satisfies the corresponding initial conditions. If we express the highest derivatives from (2) explicitly, we obtain an equation whose right-hand side satisfies the existence and uniqueness theorem. Since the operator L is linear, it means that for (2) the following holds:
1).
- solution (2), if 
- solution of inhomogeneous equation (2), and 
- solution of the corresponding homogeneous equation.
2). If 
- solutions 
, That 
solution to the equation 
.
Property 2 is the principle of superposition, it is valid when 
, if the series 
- converges and admits m-multiple term-by-term differentiation.
3) Let the operator equation be given 
, where L is an operator with coefficients 
, All 
- real. The functions U and V are also real. Then, if this equation has a solution 
, then the solution to the same equation will be both the imaginary and real parts: 
And 
. Moreover, each of them corresponds to the solution.
Theorem. General solution of the inhomogeneous equationn- about 
on the segment [a,
b] provided that all coefficients 
and right side 
- continuous functions, can be represented as the sum of the general solution corresponding to a homogeneous system 
and a particular solution to the inhomogeneous - 
.
Those. solution 
.
If it is impossible to explicitly select particular solutions of an inhomogeneous system, then you can use the method variations of constant . We will look for a solution in the form:
(3)
Where 
solutions to a homogeneous system, 
- unknown functions.
Total unknown functions 
- n. They must satisfy the original equation (2).
Substituting the expression y(x) into equation (2), we obtain conditions for determining only one unknown function. To determine the remaining (n-1)-well functions, an (n-1)-but additional condition is necessary; they can be chosen arbitrarily. Let us choose them so that the solution (2) - y(x) has the same form as if 
were constants.
,
because 
behave like constants, then 
, which means 
.
That. we get the (n-1)-but condition in addition to equation (1). If we substitute the expression for the derivatives into equation (1) and take into account all the conditions obtained and the fact that y i is the solution of the corresponding homogeneous system, then we obtain the last condition for 
.
Let's move on to the system:
(3)
The determinant of system (3) is (W) Vronsky's determinant, and because y i are solutions of a homogeneous system, then W0 on .
Example. Inhomogeneous equation
, the corresponding homogeneous equation 
We are looking for a solution in the formy= e kx . Characteristic equationk 2 +1=0, i.e.k 1,2 = i
y= e ix = cos x + i sin x, common decision -
Let's use the constant variation method:
Conditions for
:
, which is equivalent to writing: 
From here:
Linear differential systems equations.
The system of differential equations is called linear, if it is linear with respect to unknown functions and their derivatives. system n-linear equations of the 1st order are written in the form:
The system coefficients are const.
It is convenient to write this system in matrix form: ,
where is a column vector of unknown functions depending on one argument.
Column vector of derivatives of these functions.
Column vector of free members.
Coefficient matrix.
Theorem 1: If all matrix coefficients A are continuous on a certain interval and , then in a certain neighborhood of each m. 
TS&E conditions are met. Consequently, through each such point there passes a single integral curve.
Indeed, in this case, the right-hand sides of the system are continuous with respect to the set of arguments and their partial derivatives with respect to (equal to the coefficients of matrix A) are limited, due to continuity on a closed interval.
Methods for solving SLDs
1. A system of differential equations can be reduced to one equation by eliminating the unknowns.
Example: Solve the system of equations: 
(1)
Solution: exclude z from these equations. From the first equation we have . Substituting into the second equation, 
after simplification we get: 
.
This system of equations (1) reduced to a single second-order equation. After finding from this equation y, should be found z, using equality.
2. When solving a system of equations by eliminating unknowns, an equation of a higher order is usually obtained, so in many cases it is more convenient to solve the system by finding integrated combinations.
Continued 27b
Example: Solve the system 
Solution:
Let's solve this system using Euler's method. Let us write down the determinant for finding the characteristic
equation: , (since the system is homogeneous, in order for it to have a non-trivial solution, this determinant must be equal to zero). We obtain a characteristic equation and find its roots: 
The general solution is: 
;
- eigenvector.
We write down the solution for: 
;
- eigenvector.
We write down the solution for: 
;
We get the general solution: 
.
Let's check:
let's find : and substitute it into the first equation of this system, i.e. .
We get:
- true equality.
Linear diff. nth order equations. Theorem on the general solution of an inhomogeneous linear equation of the nth order.
A linear differential equation of the nth order is an equation of the form: (1)
If this equation has a coefficient, then dividing by it, we arrive at the equation: (2) .
Usually equations of the type (2). Suppose that in ur-i (2) all odds, as well as f(x) continuous on some interval (a,b). Then, according to TS&E, the equation (2) has a unique solution that satisfies the initial conditions: , , …, for . Here - any point from the interval (a,b), and all - any given numbers. The equation (2) satisfies TC&E , therefore does not have special solutions.
Def.: special the points are those at which =0.
Properties of a linear equation:
- A linear equation remains so for any change in the independent variable.
- A linear equation remains so for any linear change of the desired function.
Def: if in the equation (2) put f(x)=0, then we get an equation of the form: (3) , which is called homogeneous equation relative to the inhomogeneous equation (2).
Let us introduce the linear differential operator: (4). Using this operator, you can rewrite in short form the equation (2) And (3): L(y)=f(x), L(y)=0. Operator (4) has the following simple properties:
From these two properties a corollary can be deduced: .
Function y=y(x) is a solution to the inhomogeneous equation (2), If L(y(x))=f(x), Then f(x) called the solution to the equation. So the solution to the equation (3) called the function y(x), If L(y(x))=0 on the considered intervals.
Consider inhomogeneous linear equation: , L(y)=f(x).
Suppose that we have found a particular solution in some way, then .
Let's introduce a new unknown function z according to the formula: , where is a particular solution.
Let's substitute it into the equation: , open the brackets and get: .
The resulting equation can be rewritten as: 
Since is a particular solution to the original equation, then .
Thus, we have obtained a homogeneous equation with respect to z. The general solution to this homogeneous equation is a linear combination: , where the functions - constitute the fundamental system of solutions to the homogeneous equation. Substituting z into the replacement formula, we get: 
(*) for function y– unknown function of the original equation. All solutions to the original equation will be contained in (*).
Thus, the general solution of the inhomogeneous line. equation is represented as the sum of a general solution of a homogeneous linear equation and some particular solution of an inhomogeneous equation.
(continued on the other side)
30. Theorem of existence and uniqueness of the solution to differential. equations
Theorem: If the right side of the equation is continuous in the rectangle 
and is limited, and also satisfies the Lipschitz condition: , N=const, then there is a unique solution that satisfies the initial conditions and is defined on the segment 
, Where .
Proof:
Consider the complete metric space WITH, whose points are all possible continuous functions y(x) defined on the interval , the graphs of which lie inside the rectangle, and the distance is determined by the equality: 
. This space is often used in mathematical analysis and is called space of uniform convergence, since the convergence in the metric of this space is uniform.
Let's replace the differential. equation with given initial conditions to an equivalent integral equation: 
and consider the operator A(y), equal to the right side of this equation: 
. This operator assigns to each continuous function
Using Lipschitz's inequality, we can write that the distance . Now let’s choose one for which the following inequality would hold: .
You should choose so that , then . Thus we showed that .
According to the principle of contraction mappings, there is a single point or, what is the same, a single function - a solution to a differential equation that satisfies the given initial conditions.