x d y - y d x = \sqrt { x ^ { 2 } + y ^ { 2 } } d x
Solve for d (complex solution)
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=iy\text{ or }x=-iy\text{ or }x=0\end{matrix}\right.
Solve for d
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\end{matrix}\right.
Solve for x (complex solution)
\left\{\begin{matrix}\\x=iy\text{; }x=-iy\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.
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0=\sqrt{x^{2}+y^{2}}dx
Combine xdy and -ydx to get 0.
\sqrt{x^{2}+y^{2}}dx=0
Swap sides so that all variable terms are on the left hand side.
x\sqrt{x^{2}+y^{2}}d=0
The equation is in standard form.
d=0
Divide 0 by \sqrt{x^{2}+y^{2}}x.
0=\sqrt{x^{2}+y^{2}}dx
Combine xdy and -ydx to get 0.
\sqrt{x^{2}+y^{2}}dx=0
Swap sides so that all variable terms are on the left hand side.
x\sqrt{x^{2}+y^{2}}d=0
The equation is in standard form.
d=0
Divide 0 by \sqrt{x^{2}+y^{2}}x.
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