2.7.2.2 Example 2 \(\frac {1}{x^{5}}y^{\prime \prime }+y^{\prime }+y=0\)

\[ \frac {1}{x^{5}}y^{\prime \prime }+y^{\prime }+y=0 \]

Solved using Taylor series method.

\begin{align*} y^{\prime \prime } & =-x^{5}\left ( y^{\prime }+y\right ) \\ & =-x^{5}y-x^{5}y^{\prime }\\ y^{\prime \prime } & =f\left ( x,y,y^{\prime }\right ) \end{align*}

Hence

\[ y=y_{0}+xy_{0}^{\prime }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+1\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0},y_{0}^{\prime }}\]

Where

\begin{align*} F_{0} & =f\left ( x,y,y^{\prime }\right ) \\ F_{n} & =\frac {d}{dx}\left ( F_{n-1}\right ) \\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) F_{0}\end{align*}

Hence

\begin{align*} F_{1} & =\frac {\partial \left ( -x^{5}y-x^{5}y^{\prime }\right ) }{\partial x}+\frac {\partial \left ( -x^{5}y-x^{5}y^{\prime }\right ) }{\partial y}y^{\prime }+\frac {\partial \left ( -x^{5}y-x^{5}y^{\prime }\right ) }{\partial y^{\prime }}y^{\prime \prime }\\ & =\left ( -5x^{4}y-5x^{4}y^{\prime }\right ) -x^{5}y^{\prime }-x^{5}y^{\prime \prime }\end{align*}

But \(y^{\prime \prime }=f\left ( x,y,y^{\prime }\right ) \), the above becomes

\begin{align*} F_{1} & =\left ( -5x^{4}y-5x^{4}y^{\prime }\right ) -x^{5}y^{\prime }-x^{5}\left ( -x^{5}y-x^{5}y^{\prime }\right ) \\ & =x^{10}y-5x^{4}y-5x^{4}y^{\prime }-x^{5}y^{\prime }+x^{10}y^{\prime }\end{align*}

And

\begin{align*} F_{2} & =\frac {d}{dx}\left ( F_{n-1}\right ) \\ & =\frac {\partial }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\\ & =\frac {\partial }{\partial x}\left ( x^{10}y-5x^{4}y-5x^{4}y^{\prime }-x^{5}y^{\prime }+x^{10}y^{\prime }\right ) +\\ & +\left ( \frac {\partial }{\partial y}\left ( x^{10}y-5x^{4}y-5x^{4}y^{\prime }-x^{5}y^{\prime }+x^{10}y^{\prime }\right ) \right ) y^{\prime }\\ & +\left ( \frac {\partial }{\partial y^{\prime }}\left ( x^{10}y-5x^{4}y-5x^{4}y^{\prime }-x^{5}y^{\prime }+x^{10}y^{\prime }\right ) \right ) y^{\prime \prime }\\ & =\left ( 10x^{9}y-20x^{3}y-20x^{3}y^{\prime }-5x^{4}y^{\prime }+10x^{9}y^{\prime }\right ) +x^{4}\left ( x^{6}-5\right ) y^{\prime }+\left ( -5x^{4}-x^{5}+x^{10}\right ) y^{\prime \prime }\end{align*}

But \(y^{\prime \prime }=f\left ( x,y,y^{\prime }\right ) \), the above becomes

\begin{align*} F_{2} & =\left ( 10x^{9}y-20x^{3}y-20x^{3}y^{\prime }-5x^{4}y^{\prime }+10x^{9}y^{\prime }\right ) +x^{4}\left ( x^{6}-5\right ) y^{\prime }+\left ( -5x^{4}-x^{5}+x^{10}\right ) \left ( -x^{5}\left ( y^{\prime }+y\right ) \right ) \\ & =-x^{3}\left ( 20y+20y^{\prime }+10xy^{\prime }-15x^{6}y-x^{7}y+x^{12}y-15x^{6}y^{\prime }-2x^{7}y^{\prime }+x^{12}y^{\prime }\right ) \end{align*}

And

\begin{align*} F_{3} & =\frac {d}{dx}\left ( F_{2}\right ) \\ & =\frac {\partial }{\partial x}F_{2}+\left ( \frac {\partial F_{2}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{2}}{\partial y^{\prime }}\right ) y^{\prime \prime }\\ & =\frac {\partial }{\partial x}\left ( -x^{3}\left ( 20y+20y^{\prime }+10xy^{\prime }-15x^{6}y-x^{7}y+x^{12}y-15x^{6}y^{\prime }-2x^{7}y^{\prime }+x^{12}y^{\prime }\right ) \right ) \\ & +\left ( \frac {\partial }{\partial y}\left ( -x^{3}\left ( 20y+20y^{\prime }+10xy^{\prime }-15x^{6}y-x^{7}y+x^{12}y-15x^{6}y^{\prime }-2x^{7}y^{\prime }+x^{12}y^{\prime }\right ) \right ) \right ) y^{\prime }\\ & +\left ( \frac {\partial }{\partial y^{\prime }}\left ( -x^{3}\left ( 20y+20y^{\prime }+10xy^{\prime }-15x^{6}y-x^{7}y+x^{12}y-15x^{6}y^{\prime }-2x^{7}y^{\prime }+x^{12}y^{\prime }\right ) \right ) \right ) y^{\prime \prime }\\ & =-5x^{2}\left ( 12y+12y^{\prime }+8xy^{\prime }-27x^{6}y-2x^{7}y+3x^{12}y-27x^{6}y^{\prime }-4x^{7}y^{\prime }+3x^{12}y^{\prime }\right ) +x^{3}\left ( -x^{12}+x^{7}+15x^{6}-20\right ) y^{\prime }+\left ( -20x^{3}-10\allowbreak x^{4}+15x^{9}+2x^{10}-x^{15}\right ) y^{\prime \prime }\end{align*}

But \(y^{\prime \prime }=f\left ( x,y,y^{\prime }\right ) \), the above becomes

\begin{align*} F_{3} & =-5x^{2}\left ( 12y+12y^{\prime }+8xy^{\prime }-27x^{6}y-2x^{7}y+3x^{12}y-27x^{6}y^{\prime }-4x^{7}y^{\prime }+3x^{12}y^{\prime }\right ) +x^{3}\left ( -x^{12}+x^{7}+15x^{6}-20\right ) y^{\prime }+\left ( -20x^{3}-10\allowbreak x^{4}+15x^{9}+2x^{10}-x^{15}\right ) \left ( -x^{5}\left ( y^{\prime }+y\right ) \right ) \\ & =-x^{2}\left ( 60y+60y^{\prime }+60xy^{\prime }-155x^{6}y-20x^{7}y+30x^{12}y+2x^{13}y-x^{18}y-155x^{6}y^{\prime }-45x^{7}y^{\prime }-x^{8}y^{\prime }+30x^{12}y^{\prime }+3x^{13}y^{\prime }-x^{18}y^{\prime }\right ) \end{align*}

And so on. Since the derivatives become very complicated, the result was done on the computer which results in (Evaluating each of the above at \(x=0,y=y_{0},y^{\prime }=y_{0}^{\prime }\))

\begin{align*} F_{0} & =0\\ F_{1} & =0\\ F_{2} & =0\\ F_{3} & =0\\ F_{4} & =0\\ F_{5} & =-120y_{0}^{\prime }-120y_{0}\\ F_{6} & =-720y_{0}^{\prime }\\ F_{7} & =0\\ F_{8} & =0\\ F_{9} & =0\\ F_{10} & =0\\ F_{11} & =6652800y_{0}^{\prime }+6652800y_{0}\\ F_{12} & =79833600y_{0}^{\prime }[phone]y_{0}\\ F_{13} & =111196800y_{0}^{\prime }\\ F_{14} & =0\\ & \vdots \end{align*}

And so on. Hence

\begin{align*} y\left ( x\right ) & =y_{0}+xy_{0}^{\prime }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+2\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0},y_{0}^{\prime }}\\ & =y_{0}+xy_{0}^{\prime }+\frac {x^{7}}{7!}\left ( -120y_{0}^{\prime }-120y_{0}\right ) -\frac {x^{8}}{8!}\left ( 720y_{0}^{\prime }\right ) +\frac {x^{13}}{13!}\left ( 6652800y_{0}^{\prime }+6652800y_{0}\right ) \\ & +\frac {x^{14}}{14!}\left ( 79833600y_{0}^{\prime }[phone]y_{0}\right ) +\frac {x^{15}}{15!}\left ( 111196800y_{0}^{\prime }\right ) +\cdots \\ & =y_{0}\left ( 1-\frac {120}{7!}x^{7}+\frac {6652800}{13!}x^{13}+\frac {11404800}{14!}x^{14}-\cdots \right ) +y_{0}^{\prime }\left ( x-\frac {120}{7!}x^{7}-\frac {720}{8!}x^{8}+\frac {6652800}{13!}x^{13}+\frac {79833600}{14!}x^{14}+\frac {111196800}{15!}x^{15}+\cdots \right ) \\ & =y_{0}\left ( 1-\frac {1}{42}x^{7}+\frac {1}{936}x^{13}+\frac {1}{7644}x^{14}+\cdots \right ) +y_{0}^{\prime }\left ( x-\frac {1}{42}x^{7}-\frac {1}{56}x^{8}+\frac {1}{936}x^{13}+\frac {1}{1092}x^{14}+\frac {1}{11\,760}x^{15}+\cdots \right ) \end{align*}

Solved using power series method

Expansion around \(x=0\). This is ordinary point.  Since RHS is zero, we will find recurrence relation.

Let \(y=\sum _{n=0}^{\infty }a_{n}x^{n}\). Hence \(y^{\prime }=\sum _{n=0}^{\infty }na_{n}x^{n-1}=\sum _{n=1}^{\infty }na_{n}x^{n-1}\) and \(y^{\prime \prime }=\sum _{n=1}^{\infty }\left ( n\right ) \left ( n-1\right ) a_{n}x^{n-2}=\sum _{n=2}^{\infty }\left ( n\right ) \left ( n-1\right ) a_{n}x^{n-2}\). The ode becomes

\[ x^{-5}y^{\prime \prime }+y^{\prime }+y=0 \]

Hence

\begin{align} \sum _{n=2}^{\infty }\left ( n\right ) \left ( n-1\right ) a_{n}x^{n-2}+\sum _{n=1}^{\infty }na_{n}x^{n-1}+\sum _{n=0}^{\infty }a_{n}x^{n} & =0\nonumber \\ \sum _{n=2}^{\infty }\left ( n\right ) \left ( n-1\right ) a_{n}x^{n-7}+\sum _{n=1}^{\infty }na_{n}x^{n-1}+\sum _{n=0}^{\infty }a_{n}x^{n} & =0\nonumber \end{align}

Reindex so all powers start at lowest powers \(n-7\)

\begin{equation} \sum _{n=2}^{\infty }\left ( n\right ) \left ( n-1\right ) a_{n}x^{n-7}+\sum _{n=7}^{\infty }\left ( n-6\right ) a_{n-6}x^{n-7}+\sum _{n=7}^{\infty }a_{n-7}x^{n-7}=0 \tag {1}\end{equation}

For \(n=2,3,4,5,6\) it generates \(a_{2}=0,a_{3}=0,a_{4}=0,a_{5}=0,a_{6}=0\) since there is only one term in each one of these and the RHS is zero.

For \(n\geq 7\) we have the recurrence relation

\begin{align} \left ( n\right ) \left ( n-1\right ) a_{n}+\left ( n-6\right ) a_{n-6}+a_{n-7} & =0\tag {2}\\ a_{n} & =-\frac {\left ( n-6\right ) a_{n-6}+a_{n-7}}{\left ( n+2\right ) \left ( n+1\right ) }\nonumber \end{align}

Hence for \(n=7\)

\[ a_{7}=-\frac {a_{1}+a_{0}}{42}\]

For \(n=8\)

\[ a_{8}=-\frac {2a_{2}+a_{1}}{\left ( 6+2\right ) \left ( 6+1\right ) }=\frac {-a_{1}}{56}\]

For \(n=9\)

\[ a_{9}=-\frac {\left ( 7-4\right ) a_{3}+a_{2}}{\left ( 7+2\right ) \left ( 7+1\right ) }=0 \]

For \(n=10\)

\[ a_{10}=-\frac {\left ( 8-4\right ) a_{4}+a_{3}}{\left ( 8+2\right ) \left ( 8+1\right ) }=0 \]

For \(n=11\)

\[ a_{11}=-\frac {\left ( 9-4\right ) a_{5}+a_{4}}{\left ( 9+2\right ) \left ( 9+1\right ) }=0 \]

For \(n=12\)

\[ a_{12}=-\frac {\left ( n-4\right ) a_{6}+a_{5}}{\left ( n+2\right ) \left ( n+1\right ) }=0 \]

For \(n=13\)

\[ a_{13}=-\frac {\left ( 11-4\right ) a_{7}+a_{6}}{\left ( 11+2\right ) \left ( 11+1\right ) }=-\frac {\left ( 11-4\right ) a_{7}}{\left ( 11+2\right ) \left ( 11+1\right ) }=-\frac {7}{156}a_{7}=-\frac {7}{156}\left ( -\frac {a_{1}+a_{0}}{42}\right ) =\frac {1}{936}a_{0}+\frac {1}{936}a_{1}\]

And so on. Hence

\begin{align*} y & =\sum _{n=0}^{\infty }a_{n}x^{n}\\ & =a_{0}+a_{1}x+a_{7}x^{7}+a_{13}x^{13}+\cdots \end{align*}

Notice that all terms \(a_{n}=0\) for \(n=2\cdots 6\). The above becomes

\begin{align*} y & =a_{0}+a_{1}x+\left ( -\frac {1}{42}a_{0}-\frac {1}{42}a_{1}\right ) x^{7}+\left ( \frac {1}{936}a_{0}+\frac {1}{936}a_{1}\right ) x^{13}+\cdots \\ & =a_{0}\left ( 1-\frac {1}{42}x^{7}+\frac {1}{936}x^{13}+\cdots \right ) +a_{1}\left ( x-\frac {1}{42}x^{7}+\frac {1}{936}x^{13}+\cdots \right ) \end{align*}