Resolver (x-1)^2ydx+x^2(y+1)dy=0

$$( x - 1 ) ^ { 2 } y d x + x ^ { 2 } ( y + 1 ) d y = 0$$

Ebatzi: d (complex solution)

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

Ekuazio lineala ebazteko urratsak

\left(x-1\right)^{2} zabaltzeko, erabili Newton-en binomioa \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

$$\left(x^{2}-2x+1\right)ydx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}-2x+1 eta y biderkatzeko.

$$\left(x^{2}y-2xy+y\right)dx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}y-2xy+y eta d biderkatzeko.

$$\left(x^{2}yd-2xyd+yd\right)x+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}yd-2xyd+yd eta x biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2} eta y+1 biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}y+x^{2}\right)dy=0$$

Erabili banaketa-propietatea x^{2}y+x^{2} eta d biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}yd+x^{2}d\right)y=0$$

Erabili banaketa-propietatea x^{2}yd+x^{2}d eta y biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}dy^{2}+x^{2}dy=0$$

-ydx^{2} lortzeko, konbinatu -2ydx^{2} eta x^{2}dy.

$$ydx^{3}-ydx^{2}+ydx+x^{2}dy^{2}=0$$

Konbinatu d duten gai guztiak.

$$\left(yx^{3}-yx^{2}+yx+x^{2}y^{2}\right)d=0$$

Modu arruntean dago ekuazioa.

$$\left(x^{2}y^{2}+xy+yx^{3}-yx^{2}\right)d=0$$

Zatitu 0 balioa yx^{3}-yx^{2}+yx+x^{2}y^{2} balioarekin.

$$d=0$$

Ebatzi: d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\frac{\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\geq 3\right)\text{ or }\left(x=\frac{\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\leq -1\right)\text{ or }\left(x=\frac{-\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\geq 3\right)\text{ or }\left(x=\frac{-\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\leq -1\right)\text{ or }x=0\text{ or }y=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

\left(x-1\right)^{2} zabaltzeko, erabili Newton-en binomioa \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

$$\left(x^{2}-2x+1\right)ydx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}-2x+1 eta y biderkatzeko.

$$\left(x^{2}y-2xy+y\right)dx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}y-2xy+y eta d biderkatzeko.

$$\left(x^{2}yd-2xyd+yd\right)x+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2}yd-2xyd+yd eta x biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}\left(y+1\right)dy=0$$

Erabili banaketa-propietatea x^{2} eta y+1 biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}y+x^{2}\right)dy=0$$

Erabili banaketa-propietatea x^{2}y+x^{2} eta d biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}yd+x^{2}d\right)y=0$$

Erabili banaketa-propietatea x^{2}yd+x^{2}d eta y biderkatzeko.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}dy^{2}+x^{2}dy=0$$

-ydx^{2} lortzeko, konbinatu -2ydx^{2} eta x^{2}dy.

$$ydx^{3}-ydx^{2}+ydx+x^{2}dy^{2}=0$$

Konbinatu d duten gai guztiak.

$$\left(yx^{3}-yx^{2}+yx+x^{2}y^{2}\right)d=0$$

Modu arruntean dago ekuazioa.

$$\left(x^{2}y^{2}+xy+yx^{3}-yx^{2}\right)d=0$$

Zatitu 0 balioa yx^{3}-yx^{2}+yx+x^{2}y^{2} balioarekin.

$$d=0$$

Ebatzi: x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{; }x=0\text{; }x=\frac{\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Ebatzi: x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{; }x=\frac{-\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{, }&y\leq -1\text{ or }y\geq 3\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$