Solve ds=(v_{0}+at)dt

$$d s = ( v _ { 0 } + a t ) d t$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{tv_{0}-s}{t^{2}}\text{, }&t\neq 0\\a\in \mathrm{R}\text{, }&\left(s=0\text{ and }t=0\right)\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply v_{0}+at by d.

$$ds=\left(v_{0}d+atd\right)t$$

Use the distributive property to multiply v_{0}d+atd by t.

$$ds=v_{0}dt+adt^{2}$$

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

$$v_{0}dt+adt^{2}=ds$$

Subtract v_{0}dt from both sides.

$$adt^{2}=ds-v_{0}dt$$

Reorder the terms.

$$adt^{2}=ds-dtv_{0}$$

The equation is in standard form.

$$dt^{2}a=ds-dtv_{0}$$

Divide both sides by dt^{2}.

$$\frac{dt^{2}a}{dt^{2}}=\frac{d\left(s-tv_{0}\right)}{dt^{2}}$$

Dividing by dt^{2} undoes the multiplication by dt^{2}.

$$a=\frac{d\left(s-tv_{0}\right)}{dt^{2}}$$

Divide d\left(s-tv_{0}\right) by dt^{2}.

$$a=\frac{s-tv_{0}}{t^{2}}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&s=t\left(at+v_{0}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply v_{0}+at by d.

$$ds=\left(v_{0}d+atd\right)t$$

Use the distributive property to multiply v_{0}d+atd by t.

$$ds=v_{0}dt+adt^{2}$$

Subtract v_{0}dt from both sides.

$$ds-v_{0}dt=adt^{2}$$

Subtract adt^{2} from both sides.

$$ds-v_{0}dt-adt^{2}=0$$

Reorder the terms.

$$ds-adt^{2}-dtv_{0}=0$$

Combine all terms containing d.

$$\left(s-at^{2}-tv_{0}\right)d=0$$

Divide 0 by s-at^{2}-tv_{0}.

$$d=0$$