미분방정식 예제1 : -ydx + xdy =0

\tag{2}
\begin{align}
ydx &= xdy\\
\end{align}

\tag{3}
{1 \over x} dx = {1 \over y} dy

\tag{4}
\ln x+\ln c_1 = \ln y + \ln c_2 

\tag{5}
\begin{align}
&\ln x = \ln y + (\ln c_2 - \ln c_1)\\
&\ln x = \ln y + \ln c\\
&\ln x = \ln (cy)
\end{align}

\tag{6}
x=cy

\tag{7}
-\Big({1 \over x^2}\Big )ydx + \Big({1 \over x^2}\Big)xdy = \Big({1 \over x^2}\Big)0\\

\tag{8}
-{y \over x^2}dx + {1 \over x}dy =0

\tag{9}
{\partial f(x,y) \over \partial x} = -{y \over x^2}

\tag{10}
df(x,y) = - {y \over x^2} dx

\tag{11}
\begin{align}
\int df(x,y) = -\int {y \over x^2} dx\\
\end{align}

\tag{12}
f(x,y) = {y \over x} + k(y) = c_1

\tag{13}
{\partial f(x,y) \over \partial y} = {1 \over x}

\tag{14}
\begin{align}
{\partial \over \partial y}\Big( {{y \over x} + k(y)\Big)} = {1 \over x}\\
\cancel{1 \over x} + {{dk(y) \over {dy}}} = \cancel{1 \over x}\\
{{dk(y)\over d y}} = 0\\
\end{align}

\tag{15}
k(y) = c_2

\tag{16}
\begin{align}
f(x,y) = &{y \over x} + k(y) = c_1\\
&{y \over x} + c_2 = c_1\\
&{y \over x} = (c_1 - c_2)\\
&{y \over x} = c_3\\

\end{align}

\tag{17}
x = {y \over c_3} =cy