Примеры

Пример 2

y2uxx 2xyuxy x2uyy 0

a y2 , b xy, c x2

b2 ac 0

y2ξx xyξ y 0, yξ x xξ y 0,

dx

dy

,

y

x

xdx ydy 0, x2 y2

const,

ξ x2 y2

y2uxx 2xyuxy

x2uyy 0

y2 (4x2u 2u ) 2xy( 4xyu

2xu

)

ξξ

ξ

ξξ

ξ

x2 (4 y2u 4 yu

u

2u ) 0

ξξ

ξ

ξ

x

2u 2( y2

x2 )u 0

ξ

u

2( y2

x2 )

u

0

x2

ξ

u

2( y2

x2 )

u 0

x2

ξ

2

y

2

y , x2 ξ y2 ξ 2

ξ x

y

u

2ξ

u

0

ξ

2

ξ

Ответ:

ξ x2

y2

2ξ

u

uξ 0

ξ 2

y

ux

uξ

, uxx

uξξ

2xuξ

, uyy u

x2

(1

x2 )2

(1

x2 )2

1

(1 x2 )2 uxx uyy 2x(1 x2 )ux 0

(1 x2 )2

uξξ

2xuξ

u

2x(1 x2 )

uξ

0

2

2

2

2

2

(1

x

)

(1 x

)

1

x

uξξ u

0

Ответ:

ξ arctg x

uξξ u 0

y

n

2

u2

n

u

i

bi

cu d 0

x

i 1

x

i 1

i

i

n

2

v

n

v

εi x2 e 1x1

... n xn

2 iεi bi

e 1x1 ... n xn

x

i 1

i

i 1

i

n

2

bi i

x ... x

εi i

c ve 1 1

n n d 0

i 1