Solve dv=Tds-pdv

$$d v = T d s - p d v$$

Solve for T

$$\left\{\begin{matrix}T=\frac{v\left(p+1\right)}{s}\text{, }&s\neq 0\\T\in \mathrm{R}\text{, }&\left(v=0\text{ and }s=0\right)\text{ or }\left(p=-1\text{ and }s=0\right)\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Tds-pdv=dv$$

Add pdv to both sides.

$$Tds=dv+pdv$$

The equation is in standard form.

$$dsT=dpv+dv$$

Divide both sides by ds.

$$\frac{dsT}{ds}=\frac{dv\left(p+1\right)}{ds}$$

Dividing by ds undoes the multiplication by ds.

$$T=\frac{dv\left(p+1\right)}{ds}$$

Divide dv\left(1+p\right) by ds.

$$T=\frac{v\left(p+1\right)}{s}$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(T=0\text{ and }p=-1\right)\text{ or }\left(s=0\text{ and }p=-1\right)\text{ or }\left(v=\frac{Ts}{p+1}\text{ and }p\neq -1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Add pdv to both sides.

$$dv+pdv=Tds$$

Subtract Tds from both sides.

$$dv+pdv-Tds=0$$

Reorder the terms.

$$dpv-Tds+dv=0$$

Combine all terms containing d.

$$\left(pv-Ts+v\right)d=0$$

Divide 0 by pv-Ts+v.

$$d=0$$