Various types of differential equations with appropriate substitution will be considered in the following articles (see table 3.1).
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Table 3.1 |
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Equation of |
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Characteris- |
Substitu- |
The |
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Equation of the |
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the second |
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tics of |
tion |
second |
first order |
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order |
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equations |
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derivative |
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Dependent |
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0 |
variable y |
y ¢ = z(x) |
y ¢¢ = z ¢ |
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0 |
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F(x, y , y ) |
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absent |
F(x, z, z ) |
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Independent |
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¢ |
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0 |
variable x |
y ¢ = p( y) |
y |
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F ( y, p, p p) = 0 |
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or |
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F( y, y , y ) |
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absent |
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p p |
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Ф( y, p, p ¢) = 0 |
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Consider other types of differential equations with appropriate substitution for reduction of order:
1) a differential equation
F(x, y(k ) , y(k 1) , ..., y(n) ) 0,
which does not contain y directly and derivatives of order less than ( k 1 ) using the substitution y(k ) z(x) . After substituting, we have
F(x, z, z , ..., z (n k) ) 0 ,
where order is ( n k );
2) a differential equation
F( y, y , ..., y(n) ) 0,
which does not contain independent variable x . It can be a lesser order of times one with substituting the new function:
y = p(y),
where p(y) is a new function of y. Thereafter
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dp( y) |
dp dy |
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dy |
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y |
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y |
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d( p p) |
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d( p p) |
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2 |
) p ,… . |
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dx |
dy dx |
p p; |
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dx |
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dy |
dx |
( p p ( p ) |
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It can be expressed that the order of times one is less as well.
Consequently,
Ф( y, p, p , ..., p(n 1) ) 0 .
2.3. Linear differential equations of the order higher than the first.
The linear differential equation of the n’th order
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a (x) y(n) |
+a (x) y(n-1) |
+...+a |
n |
(x) y = f (x), |
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( 3.20 ) |
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0 |
1 |
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where a0 (x), a1 (x), ..., |
an (x), f (x) |
are the given functions, |
moreover |
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a0 (x) 0 is called linear differential equation of order n, since each term is of
the first degree in the form of y and contains its derivatives, but does not include their products.
An equation (3.20) if f (x) 0 is called right-hand member not zero. If f (x) 0 as shown,
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0 |
(x) y(n) a (x)y |
(n 1) ... a |
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(x)y 0 |
(3.21 ) |
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1 |
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it’s called homogeneous. |
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Properties of the linear homogeneous differential equation |
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1. If |
y1 is a particular |
solution |
of (3.21) then C1 y1 ( C1 = const) is a |
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particular solution of (3.21) too. |
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2. If |
y1 , y2 are the particular solutions of (3.21) then the sum of |
y1 y2 |
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and a linear combination C1 y1 C2 y2 |
are the solutions of (3.21). |
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In general case: if y1 , y2 , ..., yn are the particular solutions of (3.21) then
we are having linear combination
y C1 y1 C2 y2 ... Cn yn
is the solution of (3.21).
There are linear dependent and independent systems of the functions.
System of the functions |
y1 (x), |
y2 (x), ..., yn (x) |
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called a |
linear |
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dependent on the range (a, b), if an identity |
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1 y1 2 y2 |
... n yn 0 , |
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(3.22) |
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where 1 , 2 , ..., n are real numbers, exists if and only if |
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1 2 |
... n 0 . |
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If at least one of the terms among 1 , 2 , ..., n |
is not equal to zero and |
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the identity (3.22) exists, then |
the |
functions y1 , y2 , ..., yn are |
linear |
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For instance the functions y |
sin2 x, |
y |
2 |
cos2 x, |
y |
1 |
are linearly |
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1 |
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3 |
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dependent, when 1 1, 2 1, 3 |
1 exists an identity |
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sin 2 x cos2 x 1 0, |
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x ( ; ). |
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In order to determine whether a system of functions y1 , y2 , ..., yn is linearly dependent or it is not, the Vronsky determinant is used.
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