Internal problem ID [2984]
Internal file name [OUTPUT/2476_Sunday_June_05_2022_03_15_21_AM_73552895/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Additional problems.
Section 11.7. page 788
Problem number: 11.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Difference not integer"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}-\frac {y \left (x +1\right )}{2}=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}+\left (-\frac {x}{2}-\frac {1}{2}\right ) y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= \frac {3}{2 x}\\ q(x) &= -\frac {x +1}{2 x^{2}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0]\)
Irregular singular points : \([\infty ]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}+\left (-\frac {x}{2}-\frac {1}{2}\right ) y = 0 \] Let the solution be represented as Frobenius power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} \] Then \begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} \\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2} \\ \end{align*} Substituting the above back into the ode gives \begin{equation} \tag{1} x^{2} \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+\frac {3 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right ) x}{2}+\left (-\frac {x}{2}-\frac {1}{2}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {3 x^{n +r} a_{n} \left (n +r \right )}{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {x^{1+n +r} a_{n}}{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {a_{n} x^{n +r}}{2}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n +r\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {x^{1+n +r} a_{n}}{2}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {a_{n -1} x^{n +r}}{2}\right ) \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {3 x^{n +r} a_{n} \left (n +r \right )}{2}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {a_{n -1} x^{n +r}}{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {a_{n} x^{n +r}}{2}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {3 x^{n +r} a_{n} \left (n +r \right )}{2}-\frac {a_{n} x^{n +r}}{2} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )+\frac {3 x^{r} a_{0} r}{2}-\frac {a_{0} x^{r}}{2} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )+\frac {3 x^{r} r}{2}-\frac {x^{r}}{2}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \frac {\left (2 r^{2}+r -1\right ) x^{r}}{2} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ r^{2}+\frac {1}{2} r -\frac {1}{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {1}{2}}\\ r_2 &= -1 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \frac {\left (2 r^{2}+r -1\right ) x^{r}}{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {1}{2}}, -1\right ]\).
Since \(r_1 - r_2 = {\frac {3}{2}}\) is not an integer, then we can construct two linearly independent solutions \begin {align*} y_{1}\left (x \right ) &= x^{r_{1}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right )\\ y_{2}\left (x \right ) &= x^{r_{2}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n}\right ) \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +\frac {1}{2}}\\ y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -1} \end {align*}
We start by finding \(y_{1}\left (x \right )\). Eq (2B) derived above is now used to find all \(a_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(a_{0}\) is arbitrary and taken as \(a_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} a_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {3 a_{n} \left (n +r \right )}{2}-\frac {a_{n -1}}{2}-\frac {a_{n}}{2} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {a_{n -1}}{2 n^{2}+4 n r +2 r^{2}+n +r -1}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = \frac {a_{n -1}}{n \left (2 n +3\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {1}{2}}\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=\frac {1}{2 r^{2}+5 r +2} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{1}={\frac {1}{5}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(\frac {1}{5}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}={\frac {1}{70}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(\frac {1}{70}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {1}{1890}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(\frac {1}{70}\) |
| \(a_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(\frac {1}{1890}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}={\frac {1}{83160}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(\frac {1}{70}\) |
| \(a_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(\frac {1}{1890}\) |
| \(a_{4}\) | \(\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600}\) | \(\frac {1}{83160}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {1}{32 r^{10}+1040 r^{9}+14800 r^{8}+121160 r^{7}+629986 r^{6}+2165345 r^{5}+4955450 r^{4}+7397715 r^{3}+6810732 r^{2}+3418740 r +680400} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {1}{5405400}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(\frac {1}{70}\) |
| \(a_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(\frac {1}{1890}\) |
| \(a_{4}\) | \(\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600}\) | \(\frac {1}{83160}\) |
| \(a_{5}\) | \(\frac {1}{32 r^{10}+1040 r^{9}+14800 r^{8}+121160 r^{7}+629986 r^{6}+2165345 r^{5}+4955450 r^{4}+7397715 r^{3}+6810732 r^{2}+3418740 r +680400}\) | \(\frac {1}{5405400}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= \sqrt {x} \left (a_{0}+a_{1} x +a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}+a_{6} x^{6}\dots \right ) \\ &= \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right ) \end {align*}
Now the second solution \(y_{2}\left (x \right )\) is found. Eq (2B) derived above is now used to find all \(b_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(b_{0}\) is arbitrary and taken as \(b_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} b_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {3 b_{n} \left (n +r \right )}{2}-\frac {b_{n -1}}{2}-\frac {b_{n}}{2} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = \frac {b_{n -1}}{2 n^{2}+4 n r +2 r^{2}+n +r -1}\tag {4} \] Which for the root \(r = -1\) becomes \[ b_{n} = \frac {b_{n -1}}{n \left (2 n -3\right )}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = -1\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ b_{1}=\frac {1}{2 r^{2}+5 r +2} \] Which for the root \(r = -1\) becomes \[ b_{1}=-1 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(-1\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18} \] Which for the root \(r = -1\) becomes \[ b_{2}=-{\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(-1\) |
| \(b_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(-{\frac {1}{2}}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360} \] Which for the root \(r = -1\) becomes \[ b_{3}=-{\frac {1}{18}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(-1\) |
| \(b_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(-{\frac {1}{2}}\) |
| \(b_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(-{\frac {1}{18}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600} \] Which for the root \(r = -1\) becomes \[ b_{4}=-{\frac {1}{360}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(-1\) |
| \(b_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(-{\frac {1}{2}}\) |
| \(b_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(-{\frac {1}{18}}\) |
| \(b_{4}\) | \(\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600}\) | \(-{\frac {1}{360}}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {1}{32 r^{10}+1040 r^{9}+14800 r^{8}+121160 r^{7}+629986 r^{6}+2165345 r^{5}+4955450 r^{4}+7397715 r^{3}+6810732 r^{2}+3418740 r +680400} \] Which for the root \(r = -1\) becomes \[ b_{5}=-{\frac {1}{12600}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r^{2}+5 r +2}\) | \(-1\) |
| \(b_{2}\) | \(\frac {1}{4 r^{4}+28 r^{3}+67 r^{2}+63 r +18}\) | \(-{\frac {1}{2}}\) |
| \(b_{3}\) | \(\frac {1}{8 r^{6}+108 r^{5}+578 r^{4}+1557 r^{3}+2195 r^{2}+1494 r +360}\) | \(-{\frac {1}{18}}\) |
| \(b_{4}\) | \(\frac {1}{16 r^{8}+352 r^{7}+3272 r^{6}+16720 r^{5}+51089 r^{4}+94798 r^{3}+102943 r^{2}+58410 r +12600}\) | \(-{\frac {1}{360}}\) |
| \(b_{5}\) | \(\frac {1}{32 r^{10}+1040 r^{9}+14800 r^{8}+121160 r^{7}+629986 r^{6}+2165345 r^{5}+4955450 r^{4}+7397715 r^{3}+6810732 r^{2}+3418740 r +680400}\) | \(-{\frac {1}{12600}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= \sqrt {x} \left (b_{0}+b_{1} x +b_{2} x^{2}+b_{3} x^{3}+b_{4} x^{4}+b_{5} x^{5}+b_{6} x^{6}\dots \right ) \\ &= \frac {1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )}{x} \end {align*}
Therefore the homogeneous solution is \begin{align*} y_h(x) &= c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \\ &= c_{1} \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right ) + \frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )\right )}{x} \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )\right )}{x} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )\right )}{x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}+\left (-\frac {x}{2}-\frac {1}{2}\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {3 y^{\prime }}{2 x}+\frac {\left (x +1\right ) y}{2 x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+\frac {3 y^{\prime }}{2 x}-\frac {\left (x +1\right ) y}{2 x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {3}{2 x}, P_{3}\left (x \right )=-\frac {x +1}{2 x^{2}}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\frac {3}{2} \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\frac {1}{2} \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 x^{2} y^{\prime \prime }+3 y^{\prime } x +\left (-x -1\right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (1+r \right ) \left (-1+2 r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (k +r +1\right ) \left (2 k +2 r -1\right )-a_{k -1}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (1+r \right ) \left (-1+2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-1, \frac {1}{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 2 \left (k +r -\frac {1}{2}\right ) \left (k +r +1\right ) a_{k}-a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 2 \left (k +\frac {1}{2}+r \right ) \left (k +2+r \right ) a_{k +1}-a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k}}{\left (2 k +1+2 r \right ) \left (k +2+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-1 \\ {} & {} & a_{k +1}=\frac {a_{k}}{\left (2 k -1\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -1}, a_{k +1}=\frac {a_{k}}{\left (2 k -1\right ) \left (k +1\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +1}=\frac {a_{k}}{\left (2 k +2\right ) \left (k +\frac {5}{2}\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}}, a_{k +1}=\frac {a_{k}}{\left (2 k +2\right ) \left (k +\frac {5}{2}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -1}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {1}{2}}\right ), a_{1+k}=\frac {a_{k}}{\left (2 k -1\right ) \left (1+k \right )}, b_{1+k}=\frac {b_{k}}{\left (2 k +2\right ) \left (k +\frac {5}{2}\right )}\right ] \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)+3/2*x*diff(y(x),x)-1/2*(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 86
AsymptoticDSolveValue[x^2*y''[x]+3/2*x*y'[x]-1/2*(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{5405400}+\frac {x^4}{83160}+\frac {x^3}{1890}+\frac {x^2}{70}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {x^5}{12600}-\frac {x^4}{360}-\frac {x^3}{18}-\frac {x^2}{2}-x+1\right )}{x} \]