# Simple differential equation

zezima1
So I have the differential equation:

dV/dT = V/T

I solve it with separation and get:

ln(V) = ln(T) + c

where c has to be figured out from initial conditions.

Now this is how I am used to solving the equation. My book though does it differently. It simply integrate both sides from Vi to Vf or Ti to Tf. Why are these approaches the same? And how do you show that from the method I use?

$$\int \frac{\mathrm{d}V}{V}=\int \frac{\mathrm{d}T}{T}\\ \Rightarrow \ln(V)=\ln(T)+C_0\text{, where }C_0\text{ is just some arbitrary constant.}\\ \Rightarrow V=e^{\ln(T)+C_0}=Te^{C_0}\equiv C_1 T\text{, where }C_1\text{ is some new arbitrary constant.}$$
$$V=C_1T\text{.}$$
$$\Rightarrow V_i=C_1 T_i \Rightarrow C_1=\frac{V_i}{T_i} \Rightarrow V=\frac{V_i}{T_i}T$$
$$\Rightarrow \frac{V_f}{T_f}=\frac{V_i}{T_i}$$