4 Algorithm implementation based on modified Saunders and Smith algorithm

As with original Kovacic algorithm, an input ode

\begin{align} y^{\prime \prime }\left ( x\right ) +ay^{\prime }\left ( x\right ) +by\left ( x\right ) & =0\tag {1}\\ a,b & \in \mathbb {C} \left ( x\right ) \nonumber \end{align}

It is first transformed to the following ode by eliminating the first derivative

\begin{equation} z^{\prime \prime }=rz\qquad r\in \mathbb {C} \left ( x\right ) \tag {2}\end{equation}

Using what is known as the Liouville transformation given by

\begin{equation} y=ze^{\frac {-1}{2}\int adx} \tag {3}\end{equation}

Where it can be found that \(r\) in (2) is given by

\begin{equation} r=\frac {1}{4}a^{2}+\frac {1}{2}a^{\prime }-b \tag {4}\end{equation}

It is Eq. (2) (called DE from now on) which is solved and not Eq. (1).  There are three steps for Saunders version. Case 1 and case 2 are handled in same way. In showing the steps, both Saunders  paper (2) and Carolyn J. Smith paper (3) were used. Smith paper is more detailed and has some corrections to Saunders algorithm also.  

There are 4 steps to the algorithm. Step 0 determines which case \(r\) belongs to (case 1 or 2 or 3 or non of these). Step 1 determines the fixed \(e_{fixed},\theta _{fixed}\) and also \(e_{i},\theta _{i}\). Where \(i\) can be 0 and higher depending. (see below). Step 2 uses the \(e_{fixed},\) \(\theta _{fixed}\,\) and all the \(e_{i},\theta _{i}\) found in step 1 to determine the trial \(d,\Theta \). Here \(\Theta \) is used instead of \(\theta \) as in Smith paper so not confuse it with the \(\theta _{i}\) found in step 1.  If trial number \(d\) can be found which is integer and positive then step 3 is now called. It is in step 3 where the minimal polynomial \(p\left ( x\right ) \) is found using the \(d\) and \(\Theta \). If such \(p\left ( x\right ) \) can be found then \(\omega \) is solved for and the solution for the ode \(z^{\prime \prime }=rz\) is now \(z=e^{\int \omega dx}\). If no solution \(p\left ( x\right ) \) can be found, then the next trials \(d,\Theta \) tried in order to find \(p\left ( x\right ) \). This continues until all trials are tried or if solution is found. Below shows more details on each step. The trials \(d,\Theta \) are found by iterating over all possible set of values called \(s\). These sets of values are generated depending on case number and \(m\) value, where \(m\) is the number of terms in the square free factorization of \(t=t_{1}t_{2}^{2}t_{3}^{3}\cdots t_{m}^{m}\). How this is all done is given below in examples.