, v - v'v f(x) = 0 = 0.
v , u, .
.
. , u (x, y) , .
u (x, y)= C,
C − .
P (x, y) dx + Q (x, y) dy =0 , : ∂ Q ∂ x = ∂ P ∂ y.
- , : ∂ Q ∂ x =∂ P ∂ y.
- , u (x, y): .
- x. C , y: .
- y, u (x, y) :
.
φ (y):
.
- , φ (y) , , u (x, y):
.
- :
u (x, y)= C.
: 3, x, y. ψ (x).
.
- | - | - | - | - | - | - | - | - | - | ||
:
1) ;
2) ;
3) ;
4) .
1. .
2. .
3. .
4. .
1. . , , .
2. .
3. .
4. .
5. . .
6.
7.
15
: .
: .
|
|
.
, .
,
=f() , '=f'() "=f() , F , , ', y".
y=φ(x, C1, C2) 1 2, .
, (, , 1 2) =0, .
F(, , ', ")=0 , 1 2: =φ(, 10, 20), 10 20 .
, 1 2: (, , 10, 20)=0, 10 20 .
F(, y, ', ")=0 , 1 2. , , .
, , ()=0, '()='0. 1 2
, , (0, 0) '(x0).
.
.
q .
f()= 0, .
(2)
: 1 2 (2), 1/2 const Y=11+2y2, 1 2 , .
(2) , , , . ' ", . , , y, ' " .
= kx, k . k , (2).
' = kx (k)' = k kx, y" =k kx (kx:)'=k2 kx, , , ' " (2),
.
kx, ,
(3)
, = kx (2).
(3) :
(k1 k2) | Y1=ek1x Y2=ek2x | Y=C1ek1x+C2 ek2x | |
(k1=k2) | Y1=ek1x Y2=xek1x | Y=ek1x(C1+C2x) | |
- () | Y1=eaxcos , Y2=eaxsin | Y=eax(cos +C2 sin ) |
|
|
.
- | - | - | - | - | - | - | - | - | - | ||
1. :
2. :
.
1. .
2. .
3. .
4. .
1. .
2. .
3. .
4. .
5.
16
: .
: .
.
,
k .
. , n . n- .
. , . , .
, .
1. .
2. q , , .
3. s > 1 s ≤ 1.
. , , , .
. , . , .
. , , .
.
. , , , .
. , , .
.
1. , . , m . ( m-), .
2. S, , , A .
3. , A B , , A + B A - B .
.
, k- : .
. , , , .
.