Solve (2x+2y+3)dy-(x+y+1)dx=0

$$( 2 x + 2 y + 3 ) d y - ( x + y + 1 ) d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\frac{-\sqrt{9y^{2}+10y+1}+y-1}{2}\text{ or }x=\frac{\sqrt{9y^{2}+10y+1}+y-1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+2y+3 by d.

$$\left(2xd+2yd+3d\right)y-\left(x+y+1\right)dx=0$$

Use the distributive property to multiply 2xd+2yd+3d by y.

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

Use the distributive property to multiply x+y+1 by d.

$$2xdy+2dy^{2}+3dy-\left(xd+yd+d\right)x=0$$

Use the distributive property to multiply xd+yd+d by x.

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

To find the opposite of dx^{2}+ydx+dx, find the opposite of each term.

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

Combine 2xdy and -ydx to get xdy.

$$xdy+2dy^{2}+3dy-dx^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by xy+2y^{2}+3y-x^{2}-x.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\frac{-\sqrt{9y^{2}+10y+1}+y-1}{2}\text{ and }y\leq -1\right)\text{ or }\left(x=\frac{-\sqrt{9y^{2}+10y+1}+y-1}{2}\text{ and }y\geq -\frac{1}{9}\right)\text{ or }\left(x=\frac{\sqrt{9y^{2}+10y+1}+y-1}{2}\text{ and }y\leq -1\right)\text{ or }\left(x=\frac{\sqrt{9y^{2}+10y+1}+y-1}{2}\text{ and }y\geq -\frac{1}{9}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+2y+3 by d.

$$\left(2xd+2yd+3d\right)y-\left(x+y+1\right)dx=0$$

Use the distributive property to multiply 2xd+2yd+3d by y.

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

Use the distributive property to multiply x+y+1 by d.

$$2xdy+2dy^{2}+3dy-\left(xd+yd+d\right)x=0$$

Use the distributive property to multiply xd+yd+d by x.

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

To find the opposite of dx^{2}+ydx+dx, find the opposite of each term.

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

Combine 2xdy and -ydx to get xdy.

$$xdy+2dy^{2}+3dy-dx^{2}-dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by xy+2y^{2}+3y-x^{2}-x.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{\left(y+1\right)\left(9y+1\right)}+y-1}{2}\text{; }x=\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}+y-1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{\left(y+1\right)\left(9y+1\right)}+y-1}{2}\text{; }x=\frac{\sqrt{\left(y+1\right)\left(9y+1\right)}+y-1}{2}\text{, }&y\geq -\frac{1}{9}\text{ or }y\leq -1\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$