Today I encountered an alternate way of solving differential equations. Let's say that we have the following differential equation:
$$\frac{dP}{dt} = 0.9P$$
Step $1$: Divide both sides of equation by $P$ and this will yield
$$\frac{1}{P} \ \frac{dP}{dt} = 0.9$$
Step $2$: Replace $\frac{1}{P} \ \frac{dP}{dt}$ with $[ln(P)]'$ and this yields
$$[ln(P)]' = 0.9$$
Step $3$: We write integrals with respect to $t$ on both sides and we have
$$\int [ln(P)]'dt = \int 0.9 dt$$
Step $4$: Use $FTOC$ to integrate and we get
$$ln(P) = 0.9t + C$$
I already know that we get as our general solution $P = Ce^{0.9t}$, but why not just do separation of variables? Wouldn't that just make the work much easier than doing it this way? I would also like some clarification on Step $2$, especially where we had to replace $\frac{1}{P} \ \frac{dP}{dt}$ with $[ln(P)]'$... I can't quite grasp that just yet.