Solve 2+x/xdx+(x/y+2)dy=0

$$2 + \frac { x } { x } d x + ( \frac { x } { y } + 2 ) d y = 0$$

Solve for d

$$d=-\frac{1}{x+y}$$

$$x\neq -y\text{ and }x\neq 0\text{ and }y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by xy, the least common multiple of x,y.

$$xy\times 2+yxdx+\left(\frac{x}{y}+2\right)dyxy=0$$

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

$$xy\times 2+yx^{2}d+\left(\frac{x}{y}+2\right)dyxy=0$$

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

$$xy\times 2+yx^{2}d+\left(\frac{x}{y}+2\right)dy^{2}x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{y}{y}.

$$xy\times 2+yx^{2}d+\left(\frac{x}{y}+\frac{2y}{y}\right)dy^{2}x=0$$

Since \frac{x}{y} and \frac{2y}{y} have the same denominator, add them by adding their numerators.

$$xy\times 2+yx^{2}d+\frac{x+2y}{y}dy^{2}x=0$$

Express \frac{x+2y}{y}d as a single fraction.

$$xy\times 2+yx^{2}d+\frac{\left(x+2y\right)d}{y}y^{2}x=0$$

Express \frac{\left(x+2y\right)d}{y}y^{2} as a single fraction.

$$xy\times 2+yx^{2}d+\frac{\left(x+2y\right)dy^{2}}{y}x=0$$

Cancel out y in both numerator and denominator.

$$xy\times 2+yx^{2}d+dy\left(x+2y\right)x=0$$

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

$$xy\times 2+yx^{2}d+\left(dyx+2dy^{2}\right)x=0$$

Use the distributive property to multiply dyx+2dy^{2} by x.

$$xy\times 2+yx^{2}d+dyx^{2}+2dy^{2}x=0$$

Combine yx^{2}d and dyx^{2} to get 2yx^{2}d.

$$xy\times 2+2yx^{2}d+2dy^{2}x=0$$

Subtract xy\times 2 from both sides. Anything subtracted from zero gives its negation.

$$2yx^{2}d+2dy^{2}x=-xy\times 2$$

Multiply -1 and 2 to get -2.

$$2yx^{2}d+2dy^{2}x=-2xy$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(2xy^{2}+2yx^{2}\right)d=-2xy$$

Divide both sides by 2yx^{2}+2y^{2}x.

$$\frac{\left(2xy^{2}+2yx^{2}\right)d}{2xy^{2}+2yx^{2}}=-\frac{2xy}{2xy^{2}+2yx^{2}}$$

Dividing by 2yx^{2}+2y^{2}x undoes the multiplication by 2yx^{2}+2y^{2}x.

$$d=-\frac{2xy}{2xy^{2}+2yx^{2}}$$

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

$$d=-\frac{1}{x+y}$$

Solve for x

$$x=-y-\frac{1}{d}$$

$$d\neq -\frac{1}{y}\text{ and }d\neq 0\text{ and }y\neq 0$$