Internal problem ID [6600]
Internal file name [OUTPUT/5848_Sunday_June_05_2022_03_57_40_PM_68937364/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS
ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 17.
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
[[_Emden, _Fowler]]
\[ \boxed {4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y=0} \] 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. \[ 4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y = 0 \] 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 {1}{8 x}\\ q(x) &= \frac {1}{4 x}\\ \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 \[ 4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y = 0 \] 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} 4 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+\frac {\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )}{2}+\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 }}4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\left (n +r \right ) a_{n} x^{n +r -1}}{2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n +r -1\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r -1}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r -1} \\ \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 \(n +r -1\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\left (n +r \right ) a_{n} x^{n +r -1}}{2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {\left (n +r \right ) a_{n} x^{n +r -1}}{2} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{-1+r} a_{0} r \left (-1+r \right )+\frac {r a_{0} x^{-1+r}}{2} = 0 \] Or \[ \left (4 x^{-1+r} r \left (-1+r \right )+\frac {r \,x^{-1+r}}{2}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ r \,x^{-1+r} \left (-\frac {7}{2}+4 r \right ) = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 4 r^{2}-\frac {7}{2} r = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {7}{8}}\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ r \,x^{-1+r} \left (-\frac {7}{2}+4 r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {7}{8}}, 0\right ]\).
Since \(r_1 - r_2 = {\frac {7}{8}}\) 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 {7}{8}}\\ y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n} \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} 4 a_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {a_{n} \left (n +r \right )}{2}+a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {2 a_{n -1}}{8 n^{2}+16 n r +8 r^{2}-7 n -7 r}\tag {4} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{n} = -\frac {2 a_{n -1}}{n \left (8 n +7\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {7}{8}}\) 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 \[ a_{1}=-\frac {2}{8 r^{2}+9 r +1} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{1}=-{\frac {2}{15}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{2}={\frac {2}{345}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{3}=-{\frac {4}{32085}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
| \(a_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{32085}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{4}={\frac {2}{1251315}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
| \(a_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{32085}}\) |
| \(a_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{1251315}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{5}=-{\frac {4}{294059025}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
| \(a_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{32085}}\) |
| \(a_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{1251315}\) |
| \(a_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{294059025}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{6}={\frac {4}{48519739125}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
| \(a_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{32085}}\) |
| \(a_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{1251315}\) |
| \(a_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{294059025}}\) |
| \(a_{6}\) | \(\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )}\) | \(\frac {4}{48519739125}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=-\frac {128}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right ) \left (8 r^{2}+105 r +343\right )} \] Which for the root \(r = {\frac {7}{8}}\) becomes \[ a_{7}=-{\frac {8}{21397204954125}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-{\frac {2}{15}}\) |
| \(a_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{345}\) |
| \(a_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{32085}}\) |
| \(a_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{1251315}\) |
| \(a_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{294059025}}\) |
| \(a_{6}\) | \(\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )}\) | \(\frac {4}{48519739125}\) |
| \(a_{7}\) | \(-\frac {128}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right ) \left (8 r^{2}+105 r +343\right )}\) | \(-{\frac {8}{21397204954125}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{\frac {7}{8}} \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}+a_{7} x^{7}+a_{8} x^{8}\dots \right ) \\ &= x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\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} 4 b_{n} \left (n +r \right ) \left (n +r -1\right )+\frac {\left (n +r \right ) b_{n}}{2}+b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {2 b_{n -1}}{8 n^{2}+16 n r +8 r^{2}-7 n -7 r}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = -\frac {2 b_{n -1}}{n \left (8 n -7\right )}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = 0\) 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 \[ b_{1}=-\frac {2}{8 r^{2}+9 r +1} \] Which for the root \(r = 0\) becomes \[ b_{1}=-2 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {2}{9}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )} \] Which for the root \(r = 0\) becomes \[ b_{3}=-{\frac {4}{459}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
| \(b_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{459}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )} \] Which for the root \(r = 0\) becomes \[ b_{4}={\frac {2}{11475}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
| \(b_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{459}}\) |
| \(b_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{11475}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )} \] Which for the root \(r = 0\) becomes \[ b_{5}=-{\frac {4}{1893375}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
| \(b_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{459}}\) |
| \(b_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{11475}\) |
| \(b_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{1893375}}\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )} \] Which for the root \(r = 0\) becomes \[ b_{6}={\frac {4}{232885125}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
| \(b_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{459}}\) |
| \(b_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{11475}\) |
| \(b_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{1893375}}\) |
| \(b_{6}\) | \(\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )}\) | \(\frac {4}{232885125}\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=-\frac {128}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right ) \left (8 r^{2}+105 r +343\right )} \] Which for the root \(r = 0\) becomes \[ b_{7}=-{\frac {8}{79879597875}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{8 r^{2}+9 r +1}\) | \(-2\) |
| \(b_{2}\) | \(\frac {4}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right )}\) | \(\frac {2}{9}\) |
| \(b_{3}\) | \(-\frac {8}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right )}\) | \(-{\frac {4}{459}}\) |
| \(b_{4}\) | \(\frac {16}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right )}\) | \(\frac {2}{11475}\) |
| \(b_{5}\) | \(-\frac {32}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right )}\) | \(-{\frac {4}{1893375}}\) |
| \(b_{6}\) | \(\frac {64}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right )}\) | \(\frac {4}{232885125}\) |
| \(b_{7}\) | \(-\frac {128}{\left (8 r^{2}+9 r +1\right ) \left (8 r^{2}+25 r +18\right ) \left (8 r^{2}+41 r +51\right ) \left (8 r^{2}+57 r +100\right ) \left (8 r^{2}+73 r +165\right ) \left (8 r^{2}+89 r +246\right ) \left (8 r^{2}+105 r +343\right )}\) | \(-{\frac {8}{79879597875}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= 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}+b_{7} x^{7}+b_{8} x^{8}\dots \\ &= 1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\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 {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\right )\right ) + c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+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 {y}{4 x}-\frac {y^{\prime }}{8 x} \\ \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 {y^{\prime }}{8 x}+\frac {y}{4 x}=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 {1}{8 x}, P_{3}\left (x \right )=\frac {1}{4 x}\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 {1}{8} \\ {} & \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}}}=0 \\ {} & \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} \\ {} & {} & 8 x y^{\prime \prime }+2 y+y^{\prime }=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} y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & x \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} r \left (-7+8 r \right ) x^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +1} \left (k +1+r \right ) \left (8 k +1+8 r \right )+2 a_{k}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r \left (-7+8 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {7}{8}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 8 \left (k +\frac {1}{8}+r \right ) \left (k +1+r \right ) a_{k +1}+2 a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{\left (8 k +1+8 r \right ) \left (k +1+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{\left (8 k +1\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +1}=-\frac {2 a_{k}}{\left (8 k +1\right ) \left (k +1\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {7}{8} \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{\left (8 k +8\right ) \left (k +\frac {15}{8}\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {7}{8} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {7}{8}}, a_{k +1}=-\frac {2 a_{k}}{\left (8 k +8\right ) \left (k +\frac {15}{8}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {7}{8}}\right ), a_{k +1}=-\frac {2 a_{k}}{\left (8 k +1\right ) \left (k +1\right )}, b_{k +1}=-\frac {2 b_{k}}{\left (8 k +8\right ) \left (k +\frac {15}{8}\right )}\right ] \end {array} \]
Maple trace
`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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 52
Order:=8; dsolve(4*x*diff(y(x),x$2)+1/2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {7}{8}} \left (1-\frac {2}{15} x +\frac {2}{345} x^{2}-\frac {4}{32085} x^{3}+\frac {2}{1251315} x^{4}-\frac {4}{294059025} x^{5}+\frac {4}{48519739125} x^{6}-\frac {8}{21397204954125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2}{9} x^{2}-\frac {4}{459} x^{3}+\frac {2}{11475} x^{4}-\frac {4}{1893375} x^{5}+\frac {4}{232885125} x^{6}-\frac {8}{79879597875} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 111
AsymptoticDSolveValue[4*x*y''[x]+1/2*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {8 x^7}{79879597875}+\frac {4 x^6}{232885125}-\frac {4 x^5}{1893375}+\frac {2 x^4}{11475}-\frac {4 x^3}{459}+\frac {2 x^2}{9}-2 x+1\right )+c_1 x^{7/8} \left (-\frac {8 x^7}{21397204954125}+\frac {4 x^6}{48519739125}-\frac {4 x^5}{294059025}+\frac {2 x^4}{1251315}-\frac {4 x^3}{32085}+\frac {2 x^2}{345}-\frac {2 x}{15}+1\right ) \]