Solve du=xdx+ydy

$$d u = x d x + y d y$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$du=x^{2}d+ydy$$

Multiply y and y to get y^{2}.

$$du=x^{2}d+y^{2}d$$

Subtract x^{2}d from both sides.

$$du-x^{2}d=y^{2}d$$

Subtract y^{2}d from both sides.

$$du-x^{2}d-y^{2}d=0$$

Reorder the terms.

$$-dx^{2}-dy^{2}+du=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by u-x^{2}-y^{2}.

$$d=0$$

Solve for u (complex solution)

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

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$du=x^{2}d+ydy$$

Multiply y and y to get y^{2}.

$$du=x^{2}d+y^{2}d$$

The equation is in standard form.

$$du=dx^{2}+dy^{2}$$

Divide both sides by d.

$$\frac{du}{d}=\frac{d\left(x^{2}+y^{2}\right)}{d}$$

Dividing by d undoes the multiplication by d.

$$u=\frac{d\left(x^{2}+y^{2}\right)}{d}$$

Divide d\left(x^{2}+y^{2}\right) by d.

$$u=x^{2}+y^{2}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&u=x^{2}+y^{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$du=x^{2}d+ydy$$

Multiply y and y to get y^{2}.

$$du=x^{2}d+y^{2}d$$

Subtract x^{2}d from both sides.

$$du-x^{2}d=y^{2}d$$

Subtract y^{2}d from both sides.

$$du-x^{2}d-y^{2}d=0$$

Reorder the terms.

$$-dx^{2}-dy^{2}+du=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by -x^{2}-y^{2}+u.

$$d=0$$

Solve for u

$$\left\{\begin{matrix}\\u=x^{2}+y^{2}\text{, }&\text{unconditionally}\\u\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

$$du=x^{2}d+ydy$$

Multiply y and y to get y^{2}.

$$du=x^{2}d+y^{2}d$$

The equation is in standard form.

$$du=dx^{2}+dy^{2}$$

Divide both sides by d.

$$\frac{du}{d}=\frac{d\left(x^{2}+y^{2}\right)}{d}$$

Dividing by d undoes the multiplication by d.

$$u=\frac{d\left(x^{2}+y^{2}\right)}{d}$$

Divide d\left(x^{2}+y^{2}\right) by d.

$$u=x^{2}+y^{2}$$