Resolver (x-sqrt{xy})dy=ydx1

$$( x - \sqrt { x y } ) d y = y d x 1$$

Ebatzi: d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x_{1}=-\sqrt{xy}+x\text{ or }y=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

Erabili banaketa-propietatea x-\sqrt{xy} eta d biderkatzeko.

$$\left(xd-\sqrt{xy}d\right)y=ydx_{1}$$

Erabili banaketa-propietatea xd-\sqrt{xy}d eta y biderkatzeko.

$$xdy-\sqrt{xy}dy=ydx_{1}$$

Kendu ydx_{1} bi aldeetatik.

$$xdy-\sqrt{xy}dy-ydx_{1}=0$$

Berrantolatu gaiak.

$$dxy-dy\sqrt{xy}-dx_{1}y=0$$

Konbinatu d duten gai guztiak.

$$\left(xy-y\sqrt{xy}-x_{1}y\right)d=0$$

Modu arruntean dago ekuazioa.

$$\left(-y\sqrt{xy}+xy-x_{1}y\right)d=0$$

Zatitu 0 balioa xy-y\sqrt{xy}-x_{1}y balioarekin.

$$d=0$$

Ebatzi: d

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

Ekuazio lineala ebazteko urratsak

Erabili banaketa-propietatea x-\sqrt{xy} eta d biderkatzeko.

$$\left(xd-\sqrt{xy}d\right)y=ydx_{1}$$

Erabili banaketa-propietatea xd-\sqrt{xy}d eta y biderkatzeko.

$$xdy-\sqrt{xy}dy=ydx_{1}$$

Kendu ydx_{1} bi aldeetatik.

$$xdy-\sqrt{xy}dy-ydx_{1}=0$$

Berrantolatu gaiak.

$$dxy-dy\sqrt{xy}-dx_{1}y=0$$

Konbinatu d duten gai guztiak.

$$\left(xy-y\sqrt{xy}-x_{1}y\right)d=0$$

Modu arruntean dago ekuazioa.

$$\left(-y\sqrt{xy}+xy-x_{1}y\right)d=0$$

Zatitu 0 balioa xy-y\sqrt{xy}-x_{1}y balioarekin.

$$d=0$$

Ebatzi: x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{y\left(4x_{1}+y\right)}+y+2x_{1}}{2}\text{, }&arg(\frac{\sqrt{y\left(4x_{1}+y\right)}+y}{2})<\pi \\x=\frac{-\sqrt{y\left(4x_{1}+y\right)}+y+2x_{1}}{2}\text{, }&arg(\frac{-\sqrt{y\left(4x_{1}+y\right)}+y}{2})<\pi \\x=x_{1}\text{, }&y=0\text{ or }x_{1}=0\\x\in \mathrm{C}\text{, }&y=0\text{ or }d=0\end{matrix}\right.$$

Ebatzi: x

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