Question

$$(x-y+4)dx+(x-y+5)dy=0$$

Solve for d (complex solution)

$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\sqrt{y^{2}-5y+4}-2\text{ or }x=-\sqrt{y^{2}-5y+4}-2\end{matrix}\right.$

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Solve for d

$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\sqrt{y^{2}-5y+4}-2\text{ and }y\geq 4\right)\text{ or }\left(x=\sqrt{y^{2}-5y+4}-2\text{ and }y\leq 1\right)\text{ or }\left(x=-\sqrt{y^{2}-5y+4}-2\text{ and }y\geq 4\right)\text{ or }\left(x=-\sqrt{y^{2}-5y+4}-2\text{ and }y\leq 1\right)\end{matrix}\right.$

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Solve for x (complex solution)

$\left\{\begin{matrix}\\x=\sqrt{\left(y-4\right)\left(y-1\right)}-2\text{; }x=-\sqrt{\left(y-4\right)\left(y-1\right)}-2\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$

Solve for x

$\left\{\begin{matrix}x=\sqrt{\left(y-4\right)\left(y-1\right)}-2\text{; }x=-\sqrt{\left(y-4\right)\left(y-1\right)}-2\text{, }&y\leq 1\text{ or }y\geq 4\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$