Solve (cot^-1y+x)dy=(1+y^2)dx

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

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{C}\text{, }&x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}\text{ and }y\neq \frac{-\sqrt{3}i+1}{2}\text{ and }y\neq \frac{1+\sqrt{3}i}{2}\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \arccot(y)+x by d.

$$\left(\arccot(y)d+xd\right)y=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply \arccot(y)d+xd by y.

$$\arccot(y)dy+xdy=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply 1+y^{2} by d.

$$\arccot(y)dy+xdy=\left(d+y^{2}d\right)x$$

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

$$\arccot(y)dy+xdy=dx+y^{2}dx$$

Subtract dx from both sides.

$$\arccot(y)dy+xdy-dx=y^{2}dx$$

Subtract y^{2}dx from both sides.

$$\arccot(y)dy+xdy-dx-y^{2}dx=0$$

Reorder the terms.

$$dy\arccot(y)+dxy-dxy^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by y\arccot(y)+xy-xy^{2}-x.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}\text{, }&y\neq \frac{-\sqrt{3}i+1}{2}\text{ and }y\neq \frac{1+\sqrt{3}i}{2}\text{ and }y\neq 0\\x\in \mathrm{C}\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \arccot(y)+x by d.

$$\left(\arccot(y)d+xd\right)y=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply \arccot(y)d+xd by y.

$$\arccot(y)dy+xdy=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply 1+y^{2} by d.

$$\arccot(y)dy+xdy=\left(d+y^{2}d\right)x$$

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

$$\arccot(y)dy+xdy=dx+y^{2}dx$$

Subtract dx from both sides.

$$\arccot(y)dy+xdy-dx=y^{2}dx$$

Subtract y^{2}dx from both sides.

$$\arccot(y)dy+xdy-dx-y^{2}dx=0$$

Subtract \arccot(y)dy from both sides. Anything subtracted from zero gives its negation.

$$xdy-dx-y^{2}dx=-\arccot(y)dy$$

Combine all terms containing x.

$$\left(dy-d-y^{2}d\right)x=-\arccot(y)dy$$

The equation is in standard form.

$$\left(-dy^{2}+dy-d\right)x=-dy\arccot(y)$$

Divide both sides by dy-d-y^{2}d.

$$\frac{\left(-dy^{2}+dy-d\right)x}{-dy^{2}+dy-d}=-\frac{dy\arctan(\frac{1}{y})}{-dy^{2}+dy-d}$$

Dividing by dy-d-y^{2}d undoes the multiplication by dy-d-y^{2}d.

$$x=-\frac{dy\arctan(\frac{1}{y})}{-dy^{2}+dy-d}$$

Divide -\arctan(\frac{1}{y})dy by dy-d-y^{2}d.

$$x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \arccot(y)+x by d.

$$\left(\arccot(y)d+xd\right)y=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply \arccot(y)d+xd by y.

$$\arccot(y)dy+xdy=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply 1+y^{2} by d.

$$\arccot(y)dy+xdy=\left(d+y^{2}d\right)x$$

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

$$\arccot(y)dy+xdy=dx+y^{2}dx$$

Subtract dx from both sides.

$$\arccot(y)dy+xdy-dx=y^{2}dx$$

Subtract y^{2}dx from both sides.

$$\arccot(y)dy+xdy-dx-y^{2}dx=0$$

Reorder the terms.

$$dy\arccot(y)+dxy-dxy^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by y\arccot(y)+xy-xy^{2}-x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \arccot(y)+x by d.

$$\left(\arccot(y)d+xd\right)y=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply \arccot(y)d+xd by y.

$$\arccot(y)dy+xdy=\left(1+y^{2}\right)dx$$

Use the distributive property to multiply 1+y^{2} by d.

$$\arccot(y)dy+xdy=\left(d+y^{2}d\right)x$$

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

$$\arccot(y)dy+xdy=dx+y^{2}dx$$

Subtract dx from both sides.

$$\arccot(y)dy+xdy-dx=y^{2}dx$$

Subtract y^{2}dx from both sides.

$$\arccot(y)dy+xdy-dx-y^{2}dx=0$$

Subtract \arccot(y)dy from both sides. Anything subtracted from zero gives its negation.

$$xdy-dx-y^{2}dx=-\arccot(y)dy$$

Combine all terms containing x.

$$\left(dy-d-y^{2}d\right)x=-\arccot(y)dy$$

The equation is in standard form.

$$\left(-dy^{2}+dy-d\right)x=-dy\arccot(y)$$

Divide both sides by dy-d-y^{2}d.

$$\frac{\left(-dy^{2}+dy-d\right)x}{-dy^{2}+dy-d}=-\frac{dy\arctan(\frac{1}{y})}{-dy^{2}+dy-d}$$

Dividing by dy-d-y^{2}d undoes the multiplication by dy-d-y^{2}d.

$$x=-\frac{dy\arctan(\frac{1}{y})}{-dy^{2}+dy-d}$$

Divide -\arctan(\frac{1}{y})dy by dy-d-y^{2}d.

$$x=-\frac{y\arctan(\frac{1}{y})}{-y^{2}+y-1}$$