==========
ѿʬΥ
==========


ʬ
<tex>
\frac{dy}{dx}=f(x)g(y) \tag{1}
</tex>
ȤΤȤѿʬΥȸƤӤޤʬΤʤǰִܤȤʤǤ
ʬβˤĤơؤǤߤޤ礦


ѿʬΥβˡ
-------------------------------

򤯤ˤϤޤադƱѿˤޤȤޤĤޤѿʬΥǤ

ѿʬΥ
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 (1) ξǤϺդѿ $y$ ΤΤˡդѿ $x$ ΤˤޤȤޤ

ξդ $dx$ ݤ
<tex>
dy=f(x)g(y)dx
</tex>
Ȥʤꡤξդ $g(y)$ ǳäƤ $g(y)\ne0$ Ȥˤ
<tex>
\frac{1}{g(y)}dy = f(x)dx
</tex>
ȤʤޤϤǺѿʬΥǤޤ

ξդʬ
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Ĥξդʬޤ
<tex>
\int\frac{1}{g(y)}\,dy = \int f(x)\,dx + C \tag{2}
</tex>
 $C$ ǤʬˤǤΤȼ (2) ʬ׻
$y=$ ηˤƤСʬ򤱤פȤȤˤʤޤ
Ǥޤ༰ (2) ϰ̲ȸƤФ졤
$C$ Ͻ䶭狼ޤޤ
ʽʤɤʬʤϤΤޤäƤޤˡ



-------------------------------

Ǥ϶򤤤Ƥߤޤ礦򤯤Τ
<tex>
\frac{dy}{dx} = -\gamma y
</tex>
ȤʬǤޤѿʬΥǤ
դ $y$ 򡤱դ $x$ ޤȤޤ礦
ξդ $dx$ ݤ
<tex>
dy = -\gamma y dx
</tex>
ǡξդ $y$ ǳ
<tex>
\frac{1}{y}dy = -\gamma dx
</tex>
ȤʤäѿʬΥλǤĤξդʬޤ
<tex>
\int\frac{1}{y} dy = -\gamma \int dx + C \tag{3}
</tex>

<tex>
\int\frac{1}{y} dy = \log |y|, \quad \int dx = x
</tex>
Ǥ顤 (3) 
<tex>
\log |y| = -\gamma x + C
</tex>
ȤʤޤξդλؿȤ
<tex>
|y| &= e^{-\gamma x + C}\\
    &= e^C e^{-\gamma x}
</tex>
ä
<tex>
y = \pm e^C e^{-\gamma x}
</tex>
Ȥʤޤ $C$ ʤΤ $\pm e^C$ Ǥ顤
򿷤 $\alpha$ Ȥ֤
<tex>
y = \alpha e^{-\gamma x}
</tex>
Ȥʤޤ $y=$ ηˤʤäΤǡѿʬΥξʬ򤱤ޤ
$\alpha$ Ͻ狼ޤǤ

ꤷƤߤ
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
ĤǤʤΤǡ $x=0$ ΤȤ $y=0.5$ Ȥ
ꤷ $\alpha$ Ƥߤޤʬβ
<tex>
0.5 &= \alpha e^{-\gamma \cdot 0}\\
    &= \alpha e^{0}\\
    &= \alpha \cdot 1
</tex>
 $\alpha = 0.5$ ȵޤޤäƲꤷΤȤǤβ
<tex>
y = 0.5 e^{-\gamma x}
</tex>
Ǥ롤ȤȤˤʤޤ

Τ褦ѿʬΥʬϷޤä³ǲ򤯤ȤǤޤ
ۤˤ⤤ʷʬޤ
¿ξ硤ѷƺǽŪѿʬΥˤäƤäƲ򤯤Ȥˤʤޤ
⤦夲
<tex>
\frac{dy}{dx}=\mu(1-y)y,\quad 
\frac{dy}{dx}=-\frac{x}{y},\quad 
y^2dx-x^3dy=0
</tex>
ʤɤѿʬΥǤˤФҲ򤤤ƤߤƤ


@@author: @@
@@accept: 2004-04-28@@
