全微分方程的通解（B028）

问题

已知，

M (x, y)

d x

+

N (x, y)

d y

=

0

为全微分方程：

M (x, y)

d x

+

N (x, y)

d y

=

0

\Leftrightarrow

\frac{\partial M}{\partial y}

=

\frac{\partial N}{\partial x}

则，该全微分方程的通解 $?$

[A].

u (x, y)

=

\int_{0}^{x}

M (x, y)

d x

+

\int_{0}^{y}

N (x, y)

d y

=

C

[B].

u (x, y)

=

\int_{x}^{x_{0}}

M (x, y)

d x

+

\int_{y}^{y_{0}}

N (x, y)

d y

=

C

[C].

u (x, y)

=

\int_{x_{0}}^{x}

M (x, y)

d x

-

\int_{y_{0}}^{y}

N (x, y)

d y

=

C

[D].

u (x, y)

=

\int_{x_{0}}^{x}

M (x, y)

d x

+

\int_{y_{0}}^{y}

N (x, y)

d y

=

C

$u (x, y)$ $=$ $\int_{x_{0}}^{x}$ $M (x, y)$ $d x$ $+$ $\int_{y_{0}}^{y}$ $N (x, y)$ $d y$ $=$ $C$