Transformation of ODE of order n

We know that any differential equation (of any order) can be written in the form of a system. Therefore we will be able to solve those equations of order $n$ that can be written in a system form that we can solve, a system with constant coefficients.

Let's suppose that we have an ODE of order $n$ , $x^{n)} (t) + a_{n - 1} (t) \cdot x^{n - 1)} (t) + \dots + a_{1} (t) \cdot x^{'} (t) + a_{0} (t) \cdot x (t) = f (t)$

We define: $\begin{array}{rcl} y_{1} & = & x \\ y_{2} & = & x^{'} \\ y_{3} & = & x^{″} \\ \dots \\ y_{n} & = & x^{n - 1)} \end{array}$ Then $\begin{array}{l} y_{1}^{'} = x^{'} = y_{2} \\ y_{2}^{'} = x^{″} = y_{3} \\ \dots \\ y_{n}^{'} = x^{n)} = f (t) - a_{n - 1} (t) \cdot x^{n - 1)} (t) - \dots - a_{1} (t) \cdot x^{'} (t) - a_{0} (t) \cdot x (t) = \\ = f (t) - a_{n - 1} (t) \cdot y_{n} (t) - \dots - a_{1} (t) \cdot y_{2} (t) - a_{0} (t) \cdot y_{1} (t) \end{array}$ Obtaining a linear system of order 1. As we already said, if it is a system with constant coefficients we will be able to solve it.

Giving the solution will consist in giving the function $y_{1} (t)$ , since $x (t) = y_{1} (t)$ .

Example

For example, consider the ODE of order 5: $x^{5)} + 21 x^{3)} - 2 x^{″} - x = 2 \cos t$ We name: $\begin{array}{rcl} y_{1} & = & x \\ y_{2} & = & x^{'} \\ y_{3} & = & x^{″} \\ y_{4} & = & x^{‴} \\ y_{5} & = & x^{4)} \end{array}$

Therefore we obtain the system: $\begin{array}{l} y_{1}^{'} = y_{2} \\ y_{2}^{'} = y_{3} \\ y_{3}^{'} = y_{4} \\ y_{4}^{'} = y_{5} \\ y_{5}^{'} = - 21 y_{4} + 2 y_{3} + y_{1} + 2 \cos t \end{array}$

that we can write in matrix form: $(\begin{matrix} y_{1}^{'} \\ y_{2}^{'} \\ y_{3}^{'} \\ y_{4}^{'} \\ y_{5}^{'} \end{matrix}) = (\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 2 & - 21 & 0 \end{matrix}) (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \\ y_{5} \end{matrix}) + (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 2 \cos t \end{matrix})$

This method for obtaining a linear system is completely general. Nevertheless, bearing in mind that we can only solve systems with constant coefficients, this method will only be useful to solve ODE's of order $n$ with constant coefficients (since with this transformation a triangular system is never obtained).