Solve ∫ dD/sqrt{D}=-xintdt

$$\int \frac { d D } { \sqrt { D } } = - x \int d t$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-С-2\sqrt{D}}{t+С_{1}}\text{, }&D>0\text{ and }|t|\neq |С|\\x\in \mathrm{R}\text{, }&\left(D=С^{2}\text{ and }t=-С_{1}\text{ and }С_{2}<0\right)\text{ or }\left(С_{3}=t\text{ and }D=t^{2}\text{ and }t<0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\left(-x\right)\int 1\mathrm{d}t=\int \frac{1}{\sqrt{D}}\mathrm{d}D$$

Reorder the terms.

$$-x\int 1\mathrm{d}t=\int \frac{1}{\sqrt{D}}\mathrm{d}D$$

The equation is in standard form.

$$\left(-\left(t+С\right)\right)x=\frac{D}{\sqrt{D}}+С$$

Divide both sides by -\left(t+С\right).

$$\frac{\left(-\left(t+С\right)\right)x}{-\left(t+С\right)}=\frac{\sqrt{D}+С}{-\left(t+С\right)}$$

Dividing by -\left(t+С\right) undoes the multiplication by -\left(t+С\right).

$$x=\frac{\sqrt{D}+С}{-\left(t+С\right)}$$

Divide \sqrt{D}+С by -\left(t+С\right).

$$x=-\frac{\sqrt{D}+С}{t+С_{1}}$$