Internal problem ID [5657]
Internal file name [OUTPUT/4905_Sunday_June_05_2022_03_10_01_PM_69803307/index.tex]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT
KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.4. Bessels
Equation page 195
Problem number: 6.
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]]
\[ \boxed {x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4}=0} \] 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. \[ x^{2} y^{\prime \prime }+\left (\frac {x}{4}+\frac {3}{16}\right ) y = 0 \] 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) &= 0\\ q(x) &= \frac {4 x +3}{16 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 \[ x^{2} y^{\prime \prime }+\left (\frac {x}{4}+\frac {3}{16}\right ) y = 0 \] 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 )+\left (\frac {x}{4}+\frac {3}{16}\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 {x^{1+n +r} a_{n}}{4}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {3 a_{n} x^{n +r}}{16}\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 }}\frac {x^{1+n +r} a_{n}}{4} &= \moverset {\infty }{\munderset {n =1}{\sum }}\frac {a_{n -1} x^{n +r}}{4} \\ \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 =1}{\sum }}\frac {a_{n -1} x^{n +r}}{4}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {3 a_{n} x^{n +r}}{16}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {3 a_{n} x^{n +r}}{16} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )+\frac {3 a_{0} x^{r}}{16} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )+\frac {3 x^{r}}{16}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \frac {\left (16 r^{2}-16 r +3\right ) x^{r}}{16} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ r^{2}-r +\frac {3}{16} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {3}{4}}\\ r_2 &= {\frac {1}{4}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \frac {\left (16 r^{2}-16 r +3\right ) x^{r}}{16} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {3}{4}}, {\frac {1}{4}}\right ]\).
Since \(r_1 - r_2 = {\frac {1}{2}}\) 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 {3}{4}}\\ y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +\frac {1}{4}} \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 {a_{n -1}}{4}+\frac {3 a_{n}}{16} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {4 a_{n -1}}{16 n^{2}+32 n r +16 r^{2}-16 n -16 r +3}\tag {4} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{n} = -\frac {a_{n -1}}{4 n^{2}+2 n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {3}{4}}\) 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 \[ a_{1}=-\frac {4}{16 r^{2}+16 r +3} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{1}=-{\frac {1}{6}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{6}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{2}={\frac {1}{120}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{120}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{3}=-{\frac {1}{5040}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{120}\) |
| \(a_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{5040}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{4}={\frac {1}{362880}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{120}\) |
| \(a_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{5040}}\) |
| \(a_{4}\) | \(\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )}\) | \(\frac {1}{362880}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {1024}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right ) \left (16 r^{2}+144 r +323\right )} \] Which for the root \(r = {\frac {3}{4}}\) becomes \[ a_{5}=-{\frac {1}{39916800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{120}\) |
| \(a_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{5040}}\) |
| \(a_{4}\) | \(\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )}\) | \(\frac {1}{362880}\) |
| \(a_{5}\) | \(-\frac {1024}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right ) \left (16 r^{2}+144 r +323\right )}\) | \(-{\frac {1}{39916800}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{\frac {3}{4}} \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 ) \\ &= x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+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 {b_{n -1}}{4}+\frac {3 b_{n}}{16} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {4 b_{n -1}}{16 n^{2}+32 n r +16 r^{2}-16 n -16 r +3}\tag {4} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{n} = -\frac {b_{n -1}}{4 n^{2}-2 n}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = {\frac {1}{4}}\) 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 \[ b_{1}=-\frac {4}{16 r^{2}+16 r +3} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{1}=-{\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{2}}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{2}={\frac {1}{24}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{2}}\) |
| \(b_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{24}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{3}=-{\frac {1}{720}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{2}}\) |
| \(b_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{24}\) |
| \(b_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{720}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{4}={\frac {1}{40320}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{2}}\) |
| \(b_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{24}\) |
| \(b_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{720}}\) |
| \(b_{4}\) | \(\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )}\) | \(\frac {1}{40320}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {1024}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right ) \left (16 r^{2}+144 r +323\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{5}=-{\frac {1}{3628800}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {4}{16 r^{2}+16 r +3}\) | \(-{\frac {1}{2}}\) |
| \(b_{2}\) | \(\frac {16}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right )}\) | \(\frac {1}{24}\) |
| \(b_{3}\) | \(-\frac {64}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right )}\) | \(-{\frac {1}{720}}\) |
| \(b_{4}\) | \(\frac {256}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right )}\) | \(\frac {1}{40320}\) |
| \(b_{5}\) | \(-\frac {1024}{\left (16 r^{2}+16 r +3\right ) \left (16 r^{2}+48 r +35\right ) \left (16 r^{2}+80 r +99\right ) \left (16 r^{2}+112 r +195\right ) \left (16 r^{2}+144 r +323\right )}\) | \(-{\frac {1}{3628800}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= x^{\frac {3}{4}} \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 ) \\ &= x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \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} x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right ) + c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime \prime }+\left (\frac {x}{4}+\frac {3}{16}\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 {\left (4 x +3\right ) y}{16 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 {\left (4 x +3\right ) y}{16 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 )=0, P_{3}\left (x \right )=\frac {4 x +3}{16 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}}}=0 \\ {} & \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 {3}{16} \\ {} & \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} \\ {} & {} & 16 x^{2} y^{\prime \prime }+\left (4 x +3\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^{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+4 r \right ) \left (-3+4 r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (4 k +4 r -1\right ) \left (4 k +4 r -3\right )+4 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+4 r \right ) \left (-3+4 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {1}{4}, \frac {3}{4}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 16 \left (k +r -\frac {3}{4}\right ) \left (k +r -\frac {1}{4}\right ) a_{k}+4 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 16 \left (k +\frac {1}{4}+r \right ) \left (k +\frac {3}{4}+r \right ) a_{k +1}+4 a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {4 a_{k}}{\left (4 k +1+4 r \right ) \left (4 k +3+4 r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{4} \\ {} & {} & a_{k +1}=-\frac {4 a_{k}}{\left (4 k +2\right ) \left (4 k +4\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{4} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{4}}, a_{k +1}=-\frac {4 a_{k}}{\left (4 k +2\right ) \left (4 k +4\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {3}{4} \\ {} & {} & a_{k +1}=-\frac {4 a_{k}}{\left (4 k +4\right ) \left (4 k +6\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {3}{4} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {3}{4}}, a_{k +1}=-\frac {4 a_{k}}{\left (4 k +4\right ) \left (4 k +6\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{4}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {3}{4}}\right ), a_{k +1}=-\frac {4 a_{k}}{\left (4 k +2\right ) \left (4 k +4\right )}, b_{k +1}=-\frac {4 b_{k}}{\left (4 k +4\right ) \left (4 k +6\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)+1/4*(x+3/4)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{4}} \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 90
AsymptoticDSolveValue[x^2*y''[x]+1/4*(x+3/4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \sqrt [4]{x} \left (-\frac {x^5}{3628800}+\frac {x^4}{40320}-\frac {x^3}{720}+\frac {x^2}{24}-\frac {x}{2}+1\right )+c_1 x^{3/4} \left (-\frac {x^5}{39916800}+\frac {x^4}{362880}-\frac {x^3}{5040}+\frac {x^2}{120}-\frac {x}{6}+1\right ) \]