TM HAM Μ(X2+Y2) SAO CHO Μ(X2+Y2)(X−Y)DX+ (X+Y)DYLA VI PH^AN TOAN...

168)

Tm ham

^µ(x

²

^+y

²

⁾

sao cho

^µ(x

²

^+y

²

^{)(x−y)dx+ (x+y)dy}

la vi ph^an toan ph^an cu'a m^o.t ham

^{F(x, y)}

nao do. Tm ham

^{F(x, y)}

n^eu bi^et

^{µ(1,1) = 0;µ(√2,√2) = ln 2}HD gia’i: P(x, y) =h(x

²

+y

²

)(x−y); Q(x, y) = h(x

²

+y

²

)(x+y)

D - ^e'

^{h(x−y)dx+h(x+y)dy}

la vi ph^an toan ph^an ta pha'i co:

^∂P∂y = ∂Q

D - a.t

^{t =x}

²

^+y

²

^⇒h

⁰

^t

^{.2y(x−y)−h=h}

⁰

^t

^{.2y(x+y) +h} ∂x⇔ −h

⁰

_t

(x

²

+y

²

) = h⇔h

⁰

_t

t=h⇒h= C

₁

x

²

+y

²

.t ⇒h= C

₁

y

x

x+yx−0R⇒F(x, y) =C

1

2 +C

₁

x

²

+y

²

dy=C

1

arctgy2 ln(x

²

+y

²

) +C

2

x

²

+ 0

²

dx+C

1

0

1

F(1,1) = 0;F(√