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Initial Value Problems#

Many functions can be solutions to a given ODE on the same subset \(S\). However, we are usually interested in functions which satisfy additional properties because ODEs are ubiquitous in physics models.

Definition: Initial Value Problem

An initial value problem (IVP) consists of an \(n\)-th order ODE

\[F\left(x, y, y', y'', \dotsc, y^{(n)}\right) = 0, \]

together with a tuple \((x_0, y_0, y_1, \dotsc, y_{n-1})\) of numbers known as initial conditions.

Definition: Solution of an IVP

We say that a function \(f: \mathcal{D}_f \subseteq \mathbb{R} \to \mathbb{R}\) is a solution of the IVP on a subset \(S \subseteq \mathcal{D}_f\) if \(f\) is a solution to the ODE on \(S\) with \(x_0 \in S\) and \(f^{(k)}(x_0) = y_k\) for all \(k \in \{0, 1, \dotsc, n-1\}\).

Example: \(y'(x) - x^2 = 0\) with \(y(0) = 1\)

Consider the following initial value problem:

\[y'(x) - x^2 = 0 \qquad y(0) = 1\]

We want to find all real functions \(y\) which are solutions to the ODE \(y'(x) - x^2 = 0\) on \([0,1]\) and which also satisfy the initial condition \(y(0) = 1\).

By rearranging the ODE, we get

\[y'(x) = x^2\]

and we see that its solutions are the antiderivatives of \(x^2\). Therefore, each solution \(y\) has the form

\[y(x) = \frac{1}{3}x^3 + C\]

for some \(C \in \mathbb{R}\). Now, we need to determine specific solutions which also satisfy the initial condition:

\[y(0) = 1\]
\[\left.\left(\frac{1}{3}x^3 + C\right)\right\vert_{x = 0} = 1\]
\[C = 1\]

We can indeed verify that \(y(x) = \frac{1}{3}x^3 + 1\) satisfies both the ODE and the initial condition on \([0,1]\) and is thus a solution to the IVP. However, whether it is the only solution, is not immediately obvious.

Example: \(y'(x) = a y(x)\) with \(y(0) = y_0\)

Consider the initial value problem

\[y'(x) = a y(x) \qquad y(0) = y_0\]

The solutions of the ODE on \(\mathbb{R}\) have an exponential form

\[y(x) = c \mathrm{e}^{ax}\]

for some \(c \in \mathbb{R}\). By using the initial condition, we get a unique solution to the initial value problem:

\[y(0) = y_0\]
\[c \mathrm{e}^{a\cdot 0} = y_0\]
\[c = y_0\]
\[y(x) = y_0 \mathrm{e}^{ax}\]
Example: \(y'(t) = y^2(t)\) with \(y(0) = 1\)

Consider the following initial value problem:

\[y'(t) = y^2(t) \qquad y(0) = 1\]

On the interval \([0,1)\) it has only the following solution:

\[y(t) = \frac{1}{1-t}\]
Example: \(y''(t) + \omega^2 y(t) = 0\) with \(y(0) = y_0\) and \(y'(0) = y_1\)

Consider the initial value problem

\[y''(t) + \omega^2 y(t) = 0 \qquad y(0) = y_0 \qquad y'(0) = y_1\]

for some \(\omega, y_0, y_1 \in \mathbb{R}_{\gt 0}\). The solutions to the ODE on \(\mathbb{R}\) have the form

\[y(t) = A \cos (\omega t) + B \sin (\omega t)\]

for some \(A, B \in \mathbb{R}\). By using the initial conditions, we can find the solutions of the IVP:

\[y(0) = A \cos (\omega \cdot 0) + B \sin (\omega \cdot 0) = A\]

We thus get \(y_0 = y(0) = A\) and so \(A = y_0\). For \(y'\), we have:

\[y'(t) = -A \omega \sin (\omega t) + B \omega \cos (\omega t)\]

Using the second initial condition, we get:

\[y'(0) = -A \omega \sin (\omega \cdot 0) + B \omega \cos (\omega \cdot 0) = B \omega\]

Therefore, \(y_1 = y'(0) = B \omega\) and so \(B = \frac{y_1}{\omega}\).

The solution of the IVP on \(\mathbb{R}\) is thus the following:

\[y(t) = y_0 \cos (\omega t) + \frac{y_1}{\omega} \sin (\omega t)\]

Showing that this is the only solution on \(\mathbb{R}\) is not trivial.

Peano's Theorem: Existence of Local Solutions to Explicit IVPs

Consider the following \(n\)-th order initial value problem:

\[y^{(n)} = f(t, y, y', \dotsc, y^{(n-1)}) \qquad \begin{aligned}y(t_0) & = y_0 \\ y'(t_0) & = y_1 \\ & \vdots \\ y^{(n-1)}(t_0) & = y_{n-1} \end{aligned}\]

If \(f\) is continuous on

\[S = [t_0, t_0 + \alpha] \times [y_0 - \beta, y_0 + \beta] \times [y_1 - \beta, y_1 + \beta] \times \dotsb \times [y_{n-1} - \beta, y_{n-1} + \beta],\]

for some \(\alpha, \beta \gt 0\), then there exists a solution of the IVP on \([t_0, t_0 + \delta]\) for some \(\delta \in \left(0, \min \left\{\alpha, \frac{\beta}{M}\right\}\right]\), where

\[M = \max_{(t, p_0, p_1, \dotsc, p_{n-1}) \in S} \max (|p_1|, |p_2|, \dotsc, |p_{n-1}|, |f(t, p_0, \dotsc, p_{n-1})|).\]
Example: \(y'(t) = \sqrt{|y(t)|}\) with \(y(0) = 0\)

Consider the following initial value problem:

\[y'(t) = \sqrt{|y(t)|} \qquad y(0) = 0\]

It has an explicit first-order ODE \(y' = f(t, y)\), where \(f(t, y) = \sqrt{|y|}\), and an initial condition with \(t_0 = 0\) and \(y_0 = 0\).

For all \(\alpha, \beta \gt 0\), we see that \(f\) is continuous on

\[S = [t_0, t_0 + \alpha] \times \left\{p_0 \in \mathbb{R} : |p_0 - y_0| \le \beta \right\} = [0, \alpha] \times [-\beta, \beta]\]

and so there exists a solution of the IVP on \([t_0, t_0 + \delta] = [0, \delta]\) for some \(\delta \in \left(0, \min \left\{\alpha, \frac{\beta}{M}\right\}\right]\). Specifically,

\[M = \max_{(t, p_0) \in S} |f(t, p_0)| = \sqrt{\beta}\]

and we get:

\[\delta \in \left(0, \min \left\{\alpha, \frac{\beta}{\sqrt{\beta}}\right\}\right] = \left(0, \min \left\{\alpha, \sqrt{\beta}\right\}\right]\]

Since \(\alpha\) and \(\beta\) can be chosen to be arbitrary large, we have shown the existence of solutions for each \(\delta \gt 0\). Specifically, we proved that there is a solution of the IVP on \([0, \delta]\) for each \(\delta \gt 0\).

However, the existence of a unique solution is not guaranteed. For example, both \(y(t) = 0\) and \(y(t) = \frac{1}{4}t^2\) are solutions on each \([0, \delta]\) for each \(\delta \gt 0\).

Proof

TODO

The Theorem of Picard-Lindelöf: Existence and Uniqueness of Local Solutions to Explicit IVPs

Consider the following \(n\)-th order initial value problem:

\[y^{(n)} = f(t, y, y', \dotsc, y^{(n-1)}) \qquad \begin{aligned}y(t_0) & = y_0 \\ y'(t_0) & = y_1 \\ & \vdots \\ y^{(n-1)}(t_0) & = y_{n-1} \end{aligned}\]

If \(f\) is continuous on and satisfies a local Lipschitz condition on

\[S = [t_0, t_0 + \alpha] \times [y_0 - \beta, y_0 + \beta] \times [y_1 - \beta, y_1 + \beta] \times \dotsb \times [y_{n-1} - \beta, y_{n-1} + \beta],\]

for some \(\alpha, \beta \gt 0\), then there exists a unique solution of the IVP on \([t_0, t_0 + \delta]\) for some \(\delta \in \left(0, \min \left\{\alpha, \frac{\beta}{M}\right\}\right]\), where

\[M = \max_{(t, p_0, p_1, \dotsc, p_{n-1}) \in S} \max (|p_1|, |p_2|, \dotsc, |p_{n-1}|, |f(t, p_0, \dotsc, p_{n-1})|).\]
Proof

TODO

Theorem: Existence and Uniqueness of Solutions to Explicit IVPs

Consider the following \(n\)-th order initial value problem

\[y^{(n)} = f(t, y, y', \dotsc, y^{(n-1)}) \qquad \begin{aligned}y(t_0) & = y_0 \\ y'(t_0) & = y_1 \\ & \vdots \\ y^{(n-1)}(t_0) & = y_{n-1} \end{aligned}\]

and let \(T \gt t_0\).

If \(f\) is continuous on and satisfies a global Lipschitz condition on

\[S = [t_0, T] \times \mathbb{R}^n,\]

then there exists a unique solution of the IVP on \([t_0, T]\).

Proof

TODO