Solve dx+(2u+1/u+8)du=0

$$d x + ( \frac { 2 u + 1 } { u + 8 } ) d u = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&u\neq -8\\d\in \mathrm{C}\text{, }&x=-\frac{u\left(2u+1\right)}{u+8}\text{ and }u\neq -8\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by u+8.

$$dx\left(u+8\right)+\left(2u+1\right)du=0$$

Use the distributive property to multiply dx by u+8.

$$dxu+8dx+\left(2u+1\right)du=0$$

Use the distributive property to multiply 2u+1 by d.

$$dxu+8dx+\left(2ud+d\right)u=0$$

Use the distributive property to multiply 2ud+d by u.

$$dxu+8dx+2du^{2}+du=0$$

Combine all terms containing d.

$$\left(xu+8x+2u^{2}+u\right)d=0$$

The equation is in standard form.

$$\left(ux+8x+2u^{2}+u\right)d=0$$

Divide 0 by xu+8x+2u^{2}+u.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&u\neq -8\\d\in \mathrm{R}\text{, }&x=-\frac{u\left(2u+1\right)}{u+8}\text{ and }u\neq -8\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by u+8.

$$dx\left(u+8\right)+\left(2u+1\right)du=0$$

Use the distributive property to multiply dx by u+8.

$$dxu+8dx+\left(2u+1\right)du=0$$

Use the distributive property to multiply 2u+1 by d.

$$dxu+8dx+\left(2ud+d\right)u=0$$

Use the distributive property to multiply 2ud+d by u.

$$dxu+8dx+2du^{2}+du=0$$

Combine all terms containing d.

$$\left(xu+8x+2u^{2}+u\right)d=0$$

The equation is in standard form.

$$\left(ux+8x+2u^{2}+u\right)d=0$$

Divide 0 by xu+8x+2u^{2}+u.

$$d=0$$

Solve for u (complex solution)

$$\left\{\begin{matrix}\\u=\frac{\sqrt{x^{2}-62x+1}-x-1}{4}\text{; }u=\frac{-\sqrt{x^{2}-62x+1}-x-1}{4}\text{, }&\text{unconditionally}\\u\neq -8\text{, }&d=0\end{matrix}\right.$$

Solve for u

$$\left\{\begin{matrix}u=\frac{\sqrt{x^{2}-62x+1}-x-1}{4}\text{; }u=\frac{-\sqrt{x^{2}-62x+1}-x-1}{4}\text{, }&x\geq 8\sqrt{15}+31\text{ or }x\leq 31-8\sqrt{15}\\u\neq -8\text{, }&d=0\end{matrix}\right.$$