Direct integration initial value problems notes

Direct integration with initial value problems is the next step in complexity from direct integration ODEs. We have essentially added a constraint to the ODE by defining a point on the solution curve, $y' = f(x),\quad y(x_0)=a$.

In order to solve this differential equation, we start by finding the general solution family, as in the Direct Integration notes. We get the form $y = F(x) + C$.

Then, we use our other constraint to pin the answer: $y(x_0) = F(x_0) + C,\quad C = y(x_0) - F(x_0)$.

Notice how, given this extra constraint, we do not have a solution family unlike the first chapter. In other words, the initial value removes the freedom of the solution family.

This is a first, and very simple, look at the Picard-Lindelof Theorem, which in plain language states that there exists a unique solution to $y' = f(t,y)$ given a condition $y(t_0)=y_0$ when $\frac{\partial f}{\partial y}$ is continuous around this point. We will observe more uses of this theorem in later chapters.

There are also plenty of useful applications in this chapter. Calculus tells us that solutions of this form satisfy a function and rate-of-change relationship. For example, say we have the initial position of a runner and we are given his speed. We are able to pin where the runner ends after a certain period. This can be applied to a plethora of situations, like a leaking bucket, current from charge, and much more.