Alternate method of solving a differential equation.

Question

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.

Answer 1

Step 2: We claim that: $ \frac{1}{P} \frac{dP}{dt} = [\ln(P)]'$.

This is actually easier to prove going from right to left. Just use the chain rule and differentiate the right hand side. Let $h(x) = [\ln(x)]'$. Then

$[\ln(P)]' = h(P) * P' = \frac{1}{P} * P' = \frac{1}{P} \frac{dP}{dt}$.

No one is forcing you to do it one way or the other, but from a mathematical point of view this way is more rigorous, as it does not involve moving infinitesimals around and treats the Leibniz notation as a symbol for the derivative.

Answer 2

For this one, there probably isn't any reason to not do separation of variables. But the technique used illustrates a different way to do it, which is similar to a method used for solving first order linear differential equations.

Showing this method, could be a stepping stone for introducing that method.