Solve (x+5)dy+y^2dx=0

$$( x + 5 ) d y + y ^ { 2 } d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(x=-\frac{5}{y+1}\text{ and }y\neq -1\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+5 by d.

$$\left(xd+5d\right)y+y^{2}dx=0$$

Use the distributive property to multiply xd+5d by y.

$$xdy+5dy+y^{2}dx=0$$

Combine all terms containing d.

$$\left(xy+5y+y^{2}x\right)d=0$$

The equation is in standard form.

$$\left(xy^{2}+xy+5y\right)d=0$$

Divide 0 by xy+5y+y^{2}x.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{5}{y+1}\text{, }&y\neq -1\\x\in \mathrm{C}\text{, }&y=0\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+5 by d.

$$\left(xd+5d\right)y+y^{2}dx=0$$

Use the distributive property to multiply xd+5d by y.

$$xdy+5dy+y^{2}dx=0$$

Subtract 5dy from both sides. Anything subtracted from zero gives its negation.

$$xdy+y^{2}dx=-5dy$$

Combine all terms containing x.

$$\left(dy+y^{2}d\right)x=-5dy$$

The equation is in standard form.

$$\left(dy^{2}+dy\right)x=-5dy$$

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

$$\frac{\left(dy^{2}+dy\right)x}{dy^{2}+dy}=-\frac{5dy}{dy^{2}+dy}$$

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

$$x=-\frac{5dy}{dy^{2}+dy}$$

Divide -5dy by dy+y^{2}d.

$$x=-\frac{5}{y+1}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=-\frac{5}{y+1}\text{ and }y\neq -1\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+5 by d.

$$\left(xd+5d\right)y+y^{2}dx=0$$

Use the distributive property to multiply xd+5d by y.

$$xdy+5dy+y^{2}dx=0$$

Combine all terms containing d.

$$\left(xy+5y+y^{2}x\right)d=0$$

The equation is in standard form.

$$\left(xy^{2}+xy+5y\right)d=0$$

Divide 0 by xy+5y+y^{2}x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{5}{y+1}\text{, }&y\neq -1\\x\in \mathrm{R}\text{, }&y=0\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+5 by d.

$$\left(xd+5d\right)y+y^{2}dx=0$$

Use the distributive property to multiply xd+5d by y.

$$xdy+5dy+y^{2}dx=0$$

Subtract 5dy from both sides. Anything subtracted from zero gives its negation.

$$xdy+y^{2}dx=-5dy$$

Combine all terms containing x.

$$\left(dy+y^{2}d\right)x=-5dy$$

The equation is in standard form.

$$\left(dy^{2}+dy\right)x=-5dy$$

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

$$\frac{\left(dy^{2}+dy\right)x}{dy^{2}+dy}=-\frac{5dy}{dy^{2}+dy}$$

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

$$x=-\frac{5dy}{dy^{2}+dy}$$

Divide -5dy by dy+y^{2}d.

$$x=-\frac{5}{y+1}$$