Define the linear operator $L(y) = y'' + a_1y' + a_2y$ where $a_1,a_2 \in \Bbb{R}$. In order to show that given an initial value problem for $L(y) = 0$ the solution is unique, the following inequality is used.
Let $\phi$ be any solution of $L(y) = 0$ on an interval $\text{I}$ containing a point $x_0 \in \text{I}$. Then for all $x \in \text{I}$, the following holds, $$ ||\phi(x_0) || \exp({-k|x-x_0|)} \leq ||\phi(x)|| \leq ||\phi(x_0)|| \exp({k|x-x_0|})$$ where $||\phi(x)|| = \sqrt{|\phi(x)|^2 + |\phi^{'}(x)|^2}$ and $k = 1 + |a_1| + |a_2|$
The proof for uniqueness follows by taking two solutions $\phi ,\psi$ for the same initial value problem and noting that $\phi – \psi$ is also a solution with the initial value problem being $\phi(x_0) – \psi(x_0) = 0 , \phi'(x_0) – \psi'(x_0) = 0$ and then using the inequality mentioned above.
Question: I am unable to make sense as to how would one arrive at such a strange norm (is it even a norm ? , thinking of it as a vector in $\Bbb{R}^2$ for each point $x \in I$ it is a norm) on $\phi(x)$. It looks like the euclidean norm on the vector $(\phi(x) , \phi'(x)) \in \Bbb{R}^2$. Given the relatively less literature I have seen on Ordinary Differential Equations, I haven't seen the above norm used in many places. So how does one get the motivation to use such a norm? Are there other properties of this norm on the space of solutions of $L(y)$?
Best Answer
A common trick you'll see in ODE is to change a higher order differential equation into a system of first order equations. For example, in this case let $x_1 = y$ and $x_2 = y'$. Then we have
$$\begin{cases} x_1' = x_2 \\ x_2' = -a_2x_1 - a_1x_2 \\ \end{cases}$$
This creates an ODE of the form
$$\mathbf{x}' = \mathbf{f}(\mathbf{x}) = \mathbf{A}\mathbf{x}$$
which is both autonomous and Lipschitz in in $\mathbf{x}$. The Lipschitz condition around $y=0\implies \mathbf{x}=\mathbf{0}$ involves $||\mathbf{x}|| = \sqrt{(y)^2 + (y')^2}$, which is where that term comes from.