CALCULUS III

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KdV equation

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vdt

2

2

d xg

dt

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1 1

2 2

1 ( )'( ) ( ) ( ) ( ) ( )

'( ) ( ) ( ) ( )

x tx t a x t b y t x t

p

y t a y t b x t r y t r

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grad divdu

fdt

( ) xy f x e

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ydx

1 10

y x

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2

dy y

dx x

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2

' , (3) 11

x y yy y

y

2 2( ) 0, (1) 1x xy dy y dx y 5.

2 1 ( 2) 0y xdx x ydy

2 2' 2 4 , (2) 4xyy y x y

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22' 2 3 , (0) 5xy xy x e y

2

2 6 xdyxy xe

dx

( ) 02

y

5' 3 2 , (2) 1xy y x y

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2 3 4x xy y c Á¤ºÉ°�c�Á�}��nµ���´ªÄ�Ç

2

' , (1) 22 1

yy y

xy

M N

y x

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(2) 2 2 2( 3 ) 0x xy y dx x dy

(3) /' 2 y xxy y xe (4) 2 2' 3 2 0x y xy y (5) 2 2( 2 ) 0x y dx xy dy

­¤�µ¦Ä��n°Å��̧ÊŤnÄ�n­¤�µ¦Á�·�Á­o��(linear)(1) 2 ' 3 2 0x y xy y

(2) 2( 2 ) 0x y dx xy dy

(3) 'xy y x (4) ' 2 xxy y xe (5) 2 2( 3 ) 0x xy dx x dy

­¤�µ¦Ä��n°Å��¸ÊŤnÄ�n­¤�µ¦Á¤n��¦��(exact)

(1) 23( 2 ) (3 2 ) 02

x x dy xy y dx

(2) 2( 2 ) 2 0x y dx x dy (3) 2 2( 2 ) 0x xy dx x dy (4) 'xy y x

(5) ' 2 xxy y xe

(2 ) ( 4) 0x y dx x dy

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(1) ­¤�µ¦Â¥��´ªÂ�¦Å�o�(separable)(2) ­¤�µ¦Á°�¡´��»r�(homogeneous)(3) ­¤�µ¦Á�·�Á­o��(linear)(4) ­¤�µ¦Â¤n��¦��(exact)(5) ¤̧�o°�¼�¤µ��ªnµ�1 �o°

��®µ�¨Á�¨¥��́Â�o� (explicit solution)�°�­¤�µ¦��́�¨nµª

(1) 2 4x yx y c (2) 2

4

x cy

x

(3) 2

4

xy c

x

(4) 2 4x yx y c

(5) Ťn¤̧�ε�°��¸É�¼�

(2 ) ( 4) 0x y dx x dy

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(1) 0(2) 1(3) 4(4) 10(5) 16