Solve z(1-z^2)dx+zdy-(x+y+xz^2)dz=0

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&z=0\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply z by 1-z^{2}.

$$\left(z-z^{3}\right)dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply z-z^{3} by d.

$$\left(zd-z^{3}d\right)x+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply zd-z^{3}d by x.

$$zdx-z^{3}dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply x+y+xz^{2} by d.

$$zdx-z^{3}dx+zdy-\left(xd+yd+xz^{2}d\right)z=0$$

Use the distributive property to multiply xd+yd+xz^{2}d by z.

$$zdx-z^{3}dx+zdy-\left(xdz+ydz+xdz^{3}\right)=0$$

To find the opposite of xdz+ydz+xdz^{3}, find the opposite of each term.

$$zdx-z^{3}dx+zdy-xdz-ydz-xdz^{3}=0$$

Combine zdx and -xdz to get 0.

$$-z^{3}dx+zdy-ydz-xdz^{3}=0$$

Combine zdy and -ydz to get 0.

$$-z^{3}dx-xdz^{3}=0$$

Combine -z^{3}dx and -xdz^{3} to get -2z^{3}dx.

$$-2z^{3}dx=0$$

The equation is in standard form.

$$\left(-2xz^{3}\right)d=0$$

Divide 0 by -2z^{3}x.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply z by 1-z^{2}.

$$\left(z-z^{3}\right)dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply z-z^{3} by d.

$$\left(zd-z^{3}d\right)x+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply zd-z^{3}d by x.

$$zdx-z^{3}dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply x+y+xz^{2} by d.

$$zdx-z^{3}dx+zdy-\left(xd+yd+xz^{2}d\right)z=0$$

Use the distributive property to multiply xd+yd+xz^{2}d by z.

$$zdx-z^{3}dx+zdy-\left(xdz+ydz+xdz^{3}\right)=0$$

To find the opposite of xdz+ydz+xdz^{3}, find the opposite of each term.

$$zdx-z^{3}dx+zdy-xdz-ydz-xdz^{3}=0$$

Combine zdx and -xdz to get 0.

$$-z^{3}dx+zdy-ydz-xdz^{3}=0$$

Combine zdy and -ydz to get 0.

$$-z^{3}dx-xdz^{3}=0$$

Combine -z^{3}dx and -xdz^{3} to get -2z^{3}dx.

$$-2z^{3}dx=0$$

The equation is in standard form.

$$\left(-2dz^{3}\right)x=0$$

Divide 0 by -2z^{3}d.

$$x=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&z=0\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply z by 1-z^{2}.

$$\left(z-z^{3}\right)dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply z-z^{3} by d.

$$\left(zd-z^{3}d\right)x+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply zd-z^{3}d by x.

$$zdx-z^{3}dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply x+y+xz^{2} by d.

$$zdx-z^{3}dx+zdy-\left(xd+yd+xz^{2}d\right)z=0$$

Use the distributive property to multiply xd+yd+xz^{2}d by z.

$$zdx-z^{3}dx+zdy-\left(xdz+ydz+xdz^{3}\right)=0$$

To find the opposite of xdz+ydz+xdz^{3}, find the opposite of each term.

$$zdx-z^{3}dx+zdy-xdz-ydz-xdz^{3}=0$$

Combine zdx and -xdz to get 0.

$$-z^{3}dx+zdy-ydz-xdz^{3}=0$$

Combine zdy and -ydz to get 0.

$$-z^{3}dx-xdz^{3}=0$$

Combine -z^{3}dx and -xdz^{3} to get -2z^{3}dx.

$$-2z^{3}dx=0$$

The equation is in standard form.

$$\left(-2xz^{3}\right)d=0$$

Divide 0 by -2z^{3}x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\text{ or }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply z by 1-z^{2}.

$$\left(z-z^{3}\right)dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply z-z^{3} by d.

$$\left(zd-z^{3}d\right)x+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply zd-z^{3}d by x.

$$zdx-z^{3}dx+zdy-\left(x+y+xz^{2}\right)dz=0$$

Use the distributive property to multiply x+y+xz^{2} by d.

$$zdx-z^{3}dx+zdy-\left(xd+yd+xz^{2}d\right)z=0$$

Use the distributive property to multiply xd+yd+xz^{2}d by z.

$$zdx-z^{3}dx+zdy-\left(xdz+ydz+xdz^{3}\right)=0$$

To find the opposite of xdz+ydz+xdz^{3}, find the opposite of each term.

$$zdx-z^{3}dx+zdy-xdz-ydz-xdz^{3}=0$$

Combine zdx and -xdz to get 0.

$$-z^{3}dx+zdy-ydz-xdz^{3}=0$$

Combine zdy and -ydz to get 0.

$$-z^{3}dx-xdz^{3}=0$$

Combine -z^{3}dx and -xdz^{3} to get -2z^{3}dx.

$$-2z^{3}dx=0$$

The equation is in standard form.

$$\left(-2dz^{3}\right)x=0$$

Divide 0 by -2z^{3}d.

$$x=0$$