nth Order Ordinary Differential Equations

Definition: Let $D \subseteq \mathbb{R}^{n+1}$ be a domain (a nonempty, open, connected subset of $\mathbb{R}^{n+1}$) and let $h \in C(D, \mathbb{R})$. An nth Order Ordinary Differential Equation is of the form $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$.

The notation $y^{(i)}$ is used to the denote the $i^{\mathrm{th}}$ derivative of $y$ with respect to $t$, that is, $\displaystyle{y^{(i)} = \frac{d^iy}{dt^{i}}}$ for all $i = 0, 1, 2, ..., n$. We let $y^{(0)} = y$.

An example of a fifth order ordinary differential equation is:

(1)

\begin{align} \quad y^{(5)} = 3t + 2ty + y^{(1)} - 3y^{(4)} \end{align}

Using prime notation, the above fifth order ordinary differential equation can be written as:

(2)

\begin{align} \quad y''''' = 3t + 2ty + y' - 3y'''' \end{align}

Definition: A Solution to the nth order ordinary differential equation $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$ on an open interval $J = (a, b)$ is an n-times continuously differentiable function $\phi \in C^n(J, \mathbb{R})$ such that for all $t \in J$ we have that $(t, \phi(t), \phi^{(1)}(t), ..., \phi^{(n-1)}(t)) \in D$ and $\phi^{(n)} = h(t, \phi, \phi^{(1)}, ..., \phi^{(n-1)})$.

We can also characterize initial value problems for nth order ordinary differential equations.

Definition: An Initial Value Problem for an nth order ordinary differential equation is an nth order ODE $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$ with Initial Conditions $y^{(i-1)}(\tau) = \xi_i$ where $i = 1, 2, ..., n$ and $(\tau, \xi_1, \xi_2, ..., \xi_n) \in D$. A Solution to the initial value problem $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$ with initial conditions $y^{(i-1)}(\tau) = \xi_i$ where $i = 1, 2, ..., n$ on the open interval $J = (a, b)$ is an n-times continuously differentiable function $\phi \in C^n (J, \mathbb{R})$ such that for all $t \in J$ we have that $(t, \phi(t), \phi^{(1)}(t), ..., \phi^{(n-1)}(t)) \in D$, $\phi^{(n)} = h(t, \phi, \phi^{(1)}, ..., \phi^{(n-1)})$, and $\phi^{(i-1)}(\tau) = \xi_i$ for $i = 1, 2, ..., n$.

Theorem 1: Every initial value problem of an $n^{\mathrm{th}}$ order ordinary differential equation $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$ with initial conditions $y^{(i-1)}(\tau) = \xi_i$ where $i = 1, 2, ..., n$ on an open interval $J = (a, b)$ ($\tau \in J$) can be expressed as an initial value problem of a system of $n$ first order ordinary differential equations.

• Proof: Consider the initial value problem:

(3)

\begin{align} \quad y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)}) \quad (*) \end{align}

• With initial conditions $y^{(i-1)}(\tau) = \xi_i$ where $i = 1, 2, ..., n$ on the open interval $J = (a, b)$. We define $x_1$, $x_2$, …, $x_n$ as:

(4)

\begin{align} \quad x_1 &= y \\ \quad x_2 &= y^{(1)} \\ \quad & \vdots \\ \quad x_n &= y^{(n-1)} \end{align}

• We differentiate each of the equations above with respect to $t$ to get:

(5)

\begin{align} \quad x_1' &= y^{(1)} \\ \quad x_2' &= y^{(2)} \\ \quad & \vdots \\ \quad x_n' &= y^{(n)} \quad (**) \end{align}

• We can rewrite the above list of equations as:

(6)

\begin{align} \quad x_1' &= x_2 \\ \quad x_2' &= x_3 \\ \quad & \vdots \\ \quad x_n' &= h(t, x_1, x_2, ..., x_n) \end{align}

• The system of $n$ first order ordinary differential equations is defined for all $(t, x_1(t), x_2(t), ..., x_n(t)) \in D$.

• Now if $\phi_1$ is a solution to $(*)$ with initial conditions $\phi_1^{(i-1)} (\tau) \xi_i$ for all $i = 1, 2, ..., n$ then $\phi_1^{(n)} = h(t, \phi_1, \phi_1^{(1)}, ..., \phi_1^{(n-1)})$ and from above we have that that $\phi = (\phi_1, \phi_1^{(1)}, ..., \phi_1^{(n-1)})$ is a solution to $(**)$ satisfying $\phi(\tau) = (\xi_1, \xi_2, ..., \xi_n)$.

• Now if $\phi = (\phi_1, \phi_2, ..., \phi_n)$ is a solution to $(**)$ with initial conditions $\phi_i(\tau) = \xi_i$ for all $i = 1, 2, ..., n$ then:

(7)

\begin{align} \quad \phi_1' &= \phi_2 \\ \quad \phi_2' &= \phi_3 \\ \quad & \vdots \\ \quad \phi_n' &= h(t, \phi_1, \phi_2, ..., \phi_n) \end{align}

• From the equations above we get that:

(8)

\begin{align} \quad \phi_1^{(n)} &= h(t, \phi_1, \phi_2, ..., \phi_n) \\ &= h(t, \phi_1, \phi_1^{(1)}, ..., \phi_1^{(n-1)}) \end{align}

• So $\phi_1$ is a solution to $(*)$ and $\phi_1^{(i-1)}(\tau) = \xi_i$ for all $i = 1, 2, ..., n$. $\blacksquare$