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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}
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