Solve s(x+y)dy=dx

$$s ( x + y ) d y = d x$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply s by x+y.

$$\left(sx+sy\right)dy=dx$$

Use the distributive property to multiply sx+sy by d.

$$\left(sxd+syd\right)y=dx$$

Use the distributive property to multiply sxd+syd by y.

$$sxdy+sdy^{2}=dx$$

Subtract dx from both sides.

$$sxdy+sdy^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by sxy+sy^{2}-x.

$$d=0$$

Solve for s (complex solution)

$$\left\{\begin{matrix}s=\frac{x}{y\left(x+y\right)}\text{, }&y\neq 0\text{ and }x\neq -y\\s\in \mathrm{C}\text{, }&d=0\text{ or }\left(x=0\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply s by x+y.

$$\left(sx+sy\right)dy=dx$$

Use the distributive property to multiply sx+sy by d.

$$\left(sxd+syd\right)y=dx$$

Use the distributive property to multiply sxd+syd by y.

$$sxdy+sdy^{2}=dx$$

Combine all terms containing s.

$$\left(xdy+dy^{2}\right)s=dx$$

The equation is in standard form.

$$\left(dxy+dy^{2}\right)s=dx$$

Divide both sides by xdy+dy^{2}.

$$\frac{\left(dxy+dy^{2}\right)s}{dxy+dy^{2}}=\frac{dx}{dxy+dy^{2}}$$

Dividing by xdy+dy^{2} undoes the multiplication by xdy+dy^{2}.

$$s=\frac{dx}{dxy+dy^{2}}$$

Divide dx by xdy+dy^{2}.

$$s=\frac{x}{y\left(x+y\right)}$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply s by x+y.

$$\left(sx+sy\right)dy=dx$$

Use the distributive property to multiply sx+sy by d.

$$\left(sxd+syd\right)y=dx$$

Use the distributive property to multiply sxd+syd by y.

$$sxdy+sdy^{2}=dx$$

Subtract dx from both sides.

$$sxdy+sdy^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by sxy+sy^{2}-x.

$$d=0$$

Solve for s

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

Steps for Solving Linear Equation

Use the distributive property to multiply s by x+y.

$$\left(sx+sy\right)dy=dx$$

Use the distributive property to multiply sx+sy by d.

$$\left(sxd+syd\right)y=dx$$

Use the distributive property to multiply sxd+syd by y.

$$sxdy+sdy^{2}=dx$$

Combine all terms containing s.

$$\left(xdy+dy^{2}\right)s=dx$$

The equation is in standard form.

$$\left(dxy+dy^{2}\right)s=dx$$

Divide both sides by xdy+dy^{2}.

$$\frac{\left(dxy+dy^{2}\right)s}{dxy+dy^{2}}=\frac{dx}{dxy+dy^{2}}$$

Dividing by xdy+dy^{2} undoes the multiplication by xdy+dy^{2}.

$$s=\frac{dx}{dxy+dy^{2}}$$

Divide dx by xdy+dy^{2}.

$$s=\frac{x}{y\left(x+y\right)}$$