A unique solution exists

Existence and uniqueness

Rather than proving the existence and uniqueness of a solution, we will simply consider a theorem that states conditions under which existence and uniqueness do hold. In other words: a unique solution exists, and we’ll explore the conditions for which this is true.

Let the forcing function \(f\) be the “right-hand side” of Equation ¿eq:ioode?:

\[ f(t) \equiv b_m \frac{d^m u}{d t^m} + b_{m-1} \frac{d^{m-1} u}{d t^{m-1}} + \cdots + b_1 \frac{d u}{d t} + b_0 u. \]

Assuming this theorem can be proved, and it can (Finan2018?), we need only the initial conditions and the forcing function to guarantee ourselves there is a unique solution. Let’s think about what this means in terms of the dynamics of a system. In a sense, if we know its initial state and the input or forcing¹, how it will behave for the rest of time is determined. When I say “in a sense,” I mean that insofar as the system is well-described by Equation ¿eq:ioode?. The determinist implications of this must be understood to be approximate and limited in scope. I don’t want to be responsible for creating a bunch of determinists (Hoefer2016?).

Note however, that, given initial conditions and forcing, we only know that a unique solution exists, not what that solution is or how to find it.

Outlining a solution technique

It turns out that, given a forcing function and no initial conditions, several potential solutions can satisfy the ODE Equation ¿eq:ioode?; conversely, given certain initial conditions and no forcing function, several potential solutions satisfy the ODE. It is only when both initial conditions and a forcing function are given that a unique solution exists. It can be shown (Kreyszig2010?) that the general solution \(y_g\) (also called the total solution)—actually a “family” of solutions with unknown constants—to Equation ¿eq:ioode? is equal to the sum of two solutions that are often relatively easy to obtain:

1. the homogeneous solution \(y_h\), another family of solutions, this time to Equation ¿eq:ioode? with \(f(t) = 0\), and
2. the particular solution \(y_p\), which satisfies Equation ¿eq:ioode? sans initial conditions.

That is,

\[ y_g(t) = y_h(t) + y_p(t). \]

Methods for deriving homogeneous and particular solutions are the topics of (Section lec:homogeneous_solution?) and (Section lec:particular_solution?).

The general solution \(y_g\) is still a family of solutions that all satisfy Equation ¿eq:ioode? for a given forcing function \(f\). It only becomes the unique solution, which we call the specific solution and typically denote simply \(y\) or (occasionally) \(y_s\), once the initial conditions are applied to \(y_g\).

The diagram of figure 7.2 illustrates this solution technique, with each arrow signifying that the block at its tail is supplied to and precedes the block at its head. Lectures proceed with the diagram:

• (Section lec:homogeneous_solution?) describes how to obtain the homogeneous solution \(y_h\) from Equation ¿eq:ioode? with the forcing function \(f(t) = 0\);
• (Section lec:particular_solution?) describes how to derive the particular solution \(y_p\) from Equation ¿eq:ioode? without the initial conditions for common forcing functions by a method called undetermined coefficients;
• (Section lec:general_solution?) blows your mind by summing the homogeneous and particular solutions to obtain the general solution \(y_g\); lest we be accused of dereliction of our duty to appear smarter than Business majors, this lecture also applies the initial conditions to the general solution to find constants introduced in the homogeneous solution to fully solve the differential equation—i.e., to obtain the specific solution \(y\).