Differential Equation System

We consider solving an Nth-order single-variable explicit ODE in the following form

$$\frac{d}{dt^N}y(t)=y^{(N)}(t)=f(t, y'(t),\cdots,y^{(N-1)}(t))$$ (1)

We first convert the Nth order DE into a set of $N$ first order ODEs by defining $y_i(t)=y^{(i-1)}(t)$ ($i=1,\cdots,N$). These $N$ functions can be represented in vector form:

$${\bf y}(t)=\left[\begin{array}{c}y_1(t) y_2(t) y_3(t)\\... ...2(t) \vdots y'_{N-2}(t) y'_{N-1}(t) \end{array} \right]$$ (2)

Now the Nth-order ODE can be written as

$$\frac{d^N}{dt^N}y(t)=y^{(N)}(t) = f(t,y(t),y'(t),y''(t),\cdots,y^{(N-1)}(t))$$

$$= f(t, y_1(t),y_2(t),y_3(t),\cdots,y_N(t))=f(t, {\bf y}(t))$$ (3)

and converted into a set of $N$ first-order differential equations:

$$\frac{d}{dt}{\bf y}(t)={\bf y}'(t) =\left[\begin{array}{c}y'... ... f(t, {\bf y}(t))\end{array}\right] ={\bf f}(t, {\bf y}(t))$$ (4)

This ODE system can then be solved as a special case of an ODE system of $N$ simultaneous first-order ODEs.

In the special case of an Nth order LCCODE:

$$\sum_{n=0}^N a_n y^{(n)}(t)=x(t),\;\;\;\;\;\mbox{i.e.}\;\;\;\;\;\; y^{(N)}(t)=-\sum_{n=0}^{N-1}a_n y^{(n)}(t)+x(t)$$

where $a_N=1$. We define $y_i(t)=y^{(i-1)}(t)$ and get

$$\frac{d}{dt}{\bf y}(t)=\dot{\bf y}(t) =\left[\begin{array}... ...}{c}0 0 0 \vdots 0 x(t)\end{array}\right] \nonumber$$

which can be written in matrix form as

$$\dot{\bf y}(t)={\bf Ay}(t)+{\bf x}(t)$$