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{\bf Egzakt differenci\'alegyenletek. Integr\'al\'o t\'enyez{\H o}k}
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A
$$p(x,y)dx+q(x,y)dy$$
differenci\'alegyenlet $\mu(x,y)$ integr\'al\'o t\'enyez{\H o}je a
$$q(x,y){\partial\mu\over\partial x}- p(x,y){\partial\mu\over\partial y}
=\mu (x,y)\left({\partial p\over\partial y}- {\partial q\over\partial x}\right)
$$
parci\'alis differenci\'alegyenlet megold\'asa.
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Speci\'alis alak\'u integr\'al\'o t\'enyez{\H o}k
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Integr\'al\'o t\'enyez{\H o} & Felt\'etel & Megold\'as \\
$\mu =\mu (x)$ & ${1\over q(x,y)}
\left({\partial p\over\partial y}- {\partial q\over\partial x}\right) =f(x)$&
$\mu (x)=\exp\left(\int f(x)dx\right)$ \\
$\mu =\mu (y)$ & ${1\over p(x,y)}
\left({\partial q\over\partial x}- {\partial p\over\partial y}\right) =f(y)$&
$\mu (y)=\exp\left(\int f(y)dx\right)$ \\
$\mu =\mu (x+y)$ & ${1\over p(x,y)-q(x,y)}
\left({\partial q\over\partial x}- {\partial p\over\partial y}\right) =f(x+y)$&
$\mu (x+y)$ \\
& & $=\exp\left(\int f(x+y)d(x+y)\right)$ \\
$\mu =\mu (xy)$ & ${1\over xp(x,y)-yq(x,y)}
\left({\partial q\over\partial x}- {\partial p\over\partial y}\right) =f(xy)$&
$\mu (xy)=\exp\left(\int f(xy)d(xy)\right)$ \\
$\mu =\mu (y/x)$ & ${x^2\over xp(x,y)+yq(x,y)}
\left({\partial q\over\partial x}- {\partial p\over\partial y}\right) =f(y/x)$&
$\mu (y/x)=\exp\left(\int f(y/x)d(y/x)\right)$ \\
$\mu =\mu (x^2+y^2)$ & ${1\over 2(yp(x,y)-xq(x,y))}
\left({\partial q\over\partial x}- {\partial p\over\partial y}\right)$&
$\mu (x^2+y^2)$ \\
&$=f(x^2+y^2)$&$=\exp\left(\int f(x^2+y^2)d(x^2+y^2)\right)$ \\
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Tov\'abbi speci\'alis esetek:
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\begin{itemize}
\item Ha a differenci\'alegyenlet homog\'en foksz\'am\'u \'es
$xp(x,y)+yq(x,y)\neq 0\ ,$
akkor
$$\mu (x,y)={1\over xp(x,y)+yq(x,y)}\ .$$
\item Ha
$${\partial p\over\partial x}={\partial q\over\partial y}\ ,\
{\partial p\over\partial y}=-{\partial q\over\partial x}\ ,$$
akkor
$$\mu (x,y)={1\over p^2+q^2}\ .$$
\item Ha
$${\partial p\over\partial y}- {\partial q\over\partial x}=
p(x,y)f_1(y)-q(x,y)f_2(x)\ ,$$
akkor
$\mu (x,y)=m(x)n(y)\ ,$
ahol
$$m(x)=\exp\left(-\int f_2(x)dx\right)\ ,\
n(y)=\exp\left(-\int f_1(y)dy\right)\ .$$
\end{itemize}
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