Direct integration equations notes

Direct integration is one of the simplest types of differential equation solving, taking on the form $y' = f(x)$. One can solve these types of equations with direct integration, giving $y = \int f(x)\,dx$.

Since antiderivatives are not unique, we include a constant of integration $C$, so the solution family is $y = F(x) + C$.

This is consistent since the slope of the solution curve only depends on $x$, not $y$. As a result, the solution family only differs by a vertical shift, represented by $+C$. In essence, since we are only given the slope of $y$, we are able to recover the overall shape of the solution, but not the initial height. That comes in the next chapter.

It is important to note that solution families like this will show up everywhere in the rest of the archive. This is because differentiation is homogeneous, for example $\frac{d}{dx}[3f(x)] = 3f'(x)$, and taking derivatives kills constants. Thus, many different functions can collapse to the same derivative because constants disappear.

One interesting entry to note is $y' = e^x$ #008. Since the non-constant solution is the function itself, this function gives rise to plenty of interesting machinery to come, including integrating factors, bases for linear constant-coefficient ODEs, and much more.

Applications from this form of ODE are not super useful yet. However, we do know that from this form of equation we can recover the function-derivative relationship from a given ODE. For example, if we knew a particle's velocity, we could recover how it might move over time. Unfortunately the big restriction here is that we are not given an initial condition yet, so we cannot get an exact answer. That comes in the next chapter.