A differential expression
$\displaystyle M(x,y)dx+N(x,y)dy$
을
exact differential(완전 미분)이라고 부른다. in a region
$\displaystyle R$ of the xy-plane if
$\displaystyle \exists f\mbox{ s.t. }\frac{\partial f}{\partial x}=M(x,y)\mbox{ and }\frac{\partial f}{\partial y}=N(x,y)$
$\displaystyle f$ 는 다변수함수.
Def.
A first-order DE of the form
$\displaystyle M(x,y)dx+N(x,y)dy=0$ : exact DE
if
$\displaystyle M(x,y)dx+N(x,y)dy$ : exact differential
If
$\displaystyle z=f(x,y)$ ,
then the
전미분,total_differential is given by
$\displaystyle dz=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$
∴
$\displaystyle f(x,y)=C\qquad \Leftrightarrow\qquad \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=0$
이런 1계 미분방정식으로 표현
How can get the solution of an exact DE?
- Find $\displaystyle f(x,y)$ by
$\displaystyle f(x,y)=\int M(x,y)dx+g(y)$
or
$\displaystyle f(x,y)=\int N(x,y)dy+g(x)$
- Find $\displaystyle g(y)$ by
$\displaystyle \frac{\partial f}{\partial y}=N(x,y)$
or $\displaystyle g(x)$ by $\displaystyle \frac{\partial f}{\partial x}=M(x,y)$
⇒ Sol. $\displaystyle f(x,y)=C$ (C: const.)