Converting nth Order ODEs to Systems of n First Order ODEs

Recall from the nth Order Ordinary Differential Equations page that every $n^{\mathrm{th}}$ order ODE can be converted into a system of $n$ first order ODEs.

We will now look at some examples of doing such.

Example 1

Convert the $3^{\mathrm{rd}}$ order ODE $y''' = t + 2y + 3y''$ with initial conditions $y(0) = 1$, $y'(0) = 2$, and $y''(0) = 3$ to a system of $3$ first order ODEs.

Let:

(1)

\begin{align} \quad x_1 &= y \\ \quad x_2 &= y' \\ \quad x_3 &= y'' \end{align}

Then the equivalent system of $3$ first order ODEs is:

(2)

\begin{align} \quad x_1' &= x_2 \\ \quad x_2' &= x_3 \\ \quad x_3' &= t + 2y + 3y'' \\ &= t + 2x_1 + 3x_3 \end{align}

We now convert the initial conditions:

(3)

\begin{align} \quad y(0) = x_1(0) &= 1 \\ \quad y'(0) = x_2(0) &= 2 \\ \quad y''(0) = x_3(0) &= 3 \end{align}

Example 2

Convert the $4^{\mathrm{th}}$ order ODE $4y'''' = 3y' - ty''$ with initial conditions $y(1) = 2$, $y'(1) = 3$, $y''(1) = -6$, and $y'''(1) = 4$ to a system of $4$ first order ODEs.

Let:

(4)

\begin{align} \quad x_1 &= y \\ \quad x_2 &= y' \\ \quad x_3 &= y'' \\ \quad x_4 &= y''' \end{align}

Then:

(5)

\begin{align} \quad x_1' &= x_2 \\ \quad x_2' &= x_3 \\ \quad x_3' &= x_4 \\ \quad x_4' &= \frac{3}{4}y' - \frac{t}{4}y'' \\ &= \frac{3}{4}x_2 - \frac{t}{4} x_3 \end{align}

The initial conditions are:

(6)

\begin{align} \quad y(1) = x_1(1) &= 2 \\ \quad y'(1) = x_2(1) &= 3 \\ \quad y''(1) = x_3(1) &= -6 \\ \quad y'''(1) = x_4(1) &= 4 \end{align}