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2.4 Reduction of Higher order Equations

When we are solving initial value problems, we usually need to write these as sets of first order equations, because most of the program packages require this.

Example 1: $$ y''(x)+y(x)=0,\ y(0)=a_0,\ y'(0)=b_0 $$

We may for instance write this equation in a system as follows, $$ \begin{align*} y'(x)=&g(x)\\ g'(x)=&-y(x)\\ y(0)=&a_0,\ g(0)=b_0 \end{align*} $$

Example 2:

Another of a third order ODE is: $$ \begin{align} &y'''(x)+2y''(x)-(y'(x))^2+2y(x)=x^2 \tag{2.21}\\ &y(0)=a_0,\ y'(0)=b_0,\ y''(0)=c_0 \nonumber \end{align} $$

We set \( y'(x)=g(x) \) and \( y''(x)=g'(x)=f(x) \), and the system may be written as $$ \begin{align*} y'(x)=&g(x)\\ g'(x)=&f(x)\\ f'(x)=&-2f(x)+(g(x))^2-2y(x)+x^2 \end{align*} $$

with initial values \( y(0)=a_0,\ g(0)=b_0,\ f(0)=c_0 \).

This is fair enough for hand calculations, but when we use program packages a more systematic procedure is needed. Let's use the equation above as an example.

We start by renaming \( y \) to \( y_0 \). We then get the following procedure: $$ \begin{align*} y'&=y'_0=y_1\\ y''&=y''_0=y'_1=y_2 \end{align*} $$

Finally, the third order ODE in (2.21) may be represented as a system of first order ODEs: $$ \begin{align*} y'_0(x)=&y_1(x)\\ y'_1(x)=&y_2(x)\\ y'_2(x)=&-2y_2(x)+(y_1(x))^2-2y_0(x)+x^2 \end{align*} $$

with initial conditions \( y_0(0)=a_0,\ y_1(0)=b_0,\ y_2(0)=c_0 \).

General procedure to reduce a higher order ODE to a system of first order ODEs.

The general procedure to reduce a higher order ODE to a system of first order ODEs becomes the following:

Given the equation $$ \begin{align} y^{(m)} &=f(x,y,y',y'',\dots,y^{(m-1)}) \tag{2.22}\\ y(x_0) &=a_0, y'(x_0)=a_1, \dots,y^{(m-1)}(x_0)=a_{m-1} \nonumber \end{align} $$ where $$ \begin{align*} y^{(m)}\equiv \frac{d^my}{dx^m} \end{align*} $$ with \( y=y_0 \), we get the following system of ODEs: $$ \begin{align} y'_0 &=y_1 \nonumber \\ y'_1 &=y_2 \nonumber \\ &. \tag{2.23} \\ &.\nonumber \\ y'_{m-2}&=y_{m-1} \nonumber\\ y'_{m-1}&= f(x,y_0,y_1,y_2,\dots,y_{m-1}) \nonumber \end{align} $$ with the following boundary conditions: $$ \begin{align} y_0(x_0) &=a_0, y_1(x_0)=a_1, \dots,y_{m-1}(x_0)=a_{m-1} \nonumber \end{align} $$

2.4.1 Example: Reduction of higher order systems

Write the following ODE as a system of first order ODEs: $$ \begin{align*} y'''-y'y''-(y')^2+2y=x^3 \\ y(0)=a,\ y'(0)=b,\ y''(0)=c \end{align*} $$

First we write \( y'''=y'y''+(y')^2-2y+x^3 \).

By use of (2.23) we get $$ \begin{align*} &y_0'=y_1\\ &y_1'=y_2\\ &y_2'=y_1y_2+(y_1)^2-2y_0+x^3\\ &y_0(0)=a,\ y_1(0)=b,\ y_2=c \end{align*} $$