Internal problem ID [6949]
Internal file name [OUTPUT/6192_Friday_August_12_2022_11_04_52_PM_59914383/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots.
Exercises page 373
Problem number: 2.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Repeated root"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) y=0} \] 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. \[ 4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) y = 0 \] 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 {-1+2 x}{4 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 \[ 4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) y = 0 \] 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} 4 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 (1-2 x \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 }}4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{1+n +r} a_{n}\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 (-2 x^{1+n +r} a_{n}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 a_{n -1} x^{n +r}\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 }}4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 a_{n -1} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{r} a_{0} r \left (-1+r \right )+a_{0} x^{r} = 0 \] Or \[ \left (4 x^{r} r \left (-1+r \right )+x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (2 r -1\right )^{2} x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ \left (2 r -1\right )^{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {1}{2}}\\ r_2 &= {\frac {1}{2}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (2 r -1\right )^{2} x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {1}{2}}, {\frac {1}{2}}\right ]\).
Since the root of the indicial equation is repeated, then we can construct two linearly independent solutions. The first solution has the form \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\tag {1A} \end {align*}
Now the second solution \(y_{2}\) is found using \begin {align*} y_{2}\left (x \right ) &= y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +r}\right )\tag {1B} \end {align*}
Then the general solution will be \[ y = c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \] In Eq (1B) the sum starts from 1 and not zero. In Eq (1A), \(a_{0}\) is never zero, and is arbitrary and is typically taken as \(a_{0} = 1\), and \(\{c_{1}, c_{2}\}\) are two arbitray constants of integration which can be found from initial conditions. Using the value of the indicial root found earlier, \(r = {\frac {1}{2}}\), Eqs (1A,1B) become \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 ) &= y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +\frac {1}{2}}\right ) \end {align*}
We start by finding the first solution \(y_{1}\left (x \right )\). 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 )+a_{n}-2 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {2 a_{n -1}}{4 n^{2}+8 n r +4 r^{2}-4 n -4 r +1}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = \frac {a_{n -1}}{2 n^{2}}\tag {5} \] 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 \[ a_{1}=\frac {2}{\left (2 r +1\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{1}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}={\frac {1}{16}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {1}{288}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
| \(a_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}={\frac {1}{9216}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
| \(a_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) |
| \(a_{4}\) | \(\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}}\) | \(\frac {1}{9216}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {32}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {1}{460800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
| \(a_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) |
| \(a_{4}\) | \(\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}}\) | \(\frac {1}{9216}\) |
| \(a_{5}\) | \(\frac {32}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2}}\) | \(\frac {1}{460800}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {64}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{6}={\frac {1}{33177600}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
| \(a_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) |
| \(a_{4}\) | \(\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}}\) | \(\frac {1}{9216}\) |
| \(a_{5}\) | \(\frac {32}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2}}\) | \(\frac {1}{460800}\) |
| \(a_{6}\) | \(\frac {64}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2}}\) | \(\frac {1}{33177600}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {128}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2} \left (2 r +13\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{7}={\frac {1}{3251404800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) |
| \(a_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) |
| \(a_{4}\) | \(\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}}\) | \(\frac {1}{9216}\) |
| \(a_{5}\) | \(\frac {32}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2}}\) | \(\frac {1}{460800}\) |
| \(a_{6}\) | \(\frac {64}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2}}\) | \(\frac {1}{33177600}\) |
| \(a_{7}\) | \(\frac {128}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2} \left (2 r +13\right )^{2}}\) | \(\frac {1}{3251404800}\) |
Using the above table, then the first 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}+a_{7} x^{7}+a_{8} x^{8}\dots \right ) \\ &= \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \\ \end{align*} Now the second solution is found. The second solution is given by \[ y_{2}\left (x \right ) = y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +r}\right ) \] Where \(b_{n}\) is found using \[ b_{n} = \frac {d}{d r}a_{n ,r} \] And the above is then evaluated at \(r = {\frac {1}{2}}\). The above table for \(a_{n ,r}\) is used for this purpose. Computing the derivatives gives the following table
| \(n\) | \(b_{n ,r}\) | \(a_{n}\) | \(b_{n ,r} = \frac {d}{d r}a_{n ,r}\) | \(b_{n}\left (r =\frac {1}{2}\right )\) |
| \(b_{0}\) | \(1\) | \(1\) | N/A since \(b_{n}\) starts from 1 | N/A |
| \(b_{1}\) | \(\frac {2}{\left (2 r +1\right )^{2}}\) | \(\frac {1}{2}\) | \(-\frac {8}{\left (2 r +1\right )^{3}}\) | \(-1\) |
| \(b_{2}\) | \(\frac {4}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2}}\) | \(\frac {1}{16}\) | \(\frac {-64 r -64}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3}}\) | \(-{\frac {3}{16}}\) |
| \(b_{3}\) | \(\frac {8}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2}}\) | \(\frac {1}{288}\) | \(\frac {-384 r^{2}-1152 r -736}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3} \left (2 r +5\right )^{3}}\) | \(-{\frac {11}{864}}\) |
| \(b_{4}\) | \(\frac {16}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2}}\) | \(\frac {1}{9216}\) | \(-\frac {512 \left (4 r^{3}+24 r^{2}+43 r +22\right )}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3}}\) | \(-{\frac {25}{55296}}\) |
| \(b_{5}\) | \(\frac {32}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2}}\) | \(\frac {1}{460800}\) | \(-\frac {128 \left (80 r^{4}+800 r^{3}+2760 r^{2}+3800 r +1689\right )}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3} \left (2 r +9\right )^{3}}\) | \(-{\frac {137}{13824000}}\) |
| \(b_{6}\) | \(\frac {64}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2}}\) | \(\frac {1}{33177600}\) | \(-\frac {49152 \left (r +3\right ) \left (r^{4}+12 r^{3}+\frac {289}{6} r^{2}+73 r +\frac {1627}{48}\right )}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3} \left (2 r +9\right )^{3} \left (2 r +11\right )^{3}}\) | \(-{\frac {49}{331776000}}\) |
| \(b_{7}\) | \(\frac {128}{\left (2 r +1\right )^{2} \left (2 r +3\right )^{2} \left (2 r +5\right )^{2} \left (2 r +7\right )^{2} \left (2 r +9\right )^{2} \left (2 r +11\right )^{2} \left (2 r +13\right )^{2}}\) | \(\frac {1}{3251404800}\) | \(-\frac {512 \left (448 r^{6}+9408 r^{5}+77840 r^{4}+321440 r^{3}+688548 r^{2}+709324 r +264207\right )}{\left (2 r +1\right )^{3} \left (2 r +3\right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3} \left (2 r +9\right )^{3} \left (2 r +11\right )^{3} \left (2 r +13\right )^{3}}\) | \(-{\frac {121}{75866112000}}\) |
The above table gives all values of \(b_{n}\) needed. Hence the second solution is \begin{align*} y_{2}\left (x \right )&=y_{1}\left (x \right ) \ln \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 \\ &= \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\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} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) + c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\right )\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\right )\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\right )\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (1-2 x \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\left (-1+2 x \right ) y}{4 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} \\ {} & {} & \frac {d}{d x}y^{\prime }-\frac {\left (-1+2 x \right ) y}{4 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 {-1+2 x}{4 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 {1}{4} \\ {} & \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} \\ {} & {} & 4 x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (1-2 x \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 \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\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+2 r \right )^{2} x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (2 k +2 r -1\right )^{2}-2 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+2 r \right )^{2}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r =\frac {1}{2} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 4 \left (k +r -\frac {1}{2}\right )^{2} a_{k}-2 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 4 \left (k +\frac {1}{2}+r \right )^{2} 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 (2 k +1+2 r \right )^{2}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +1}=\frac {2 a_{k}}{\left (2 k +2\right )^{2}} \\ \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 {2 a_{k}}{\left (2 k +2\right )^{2}}\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: 0.015 (sec). Leaf size: 81
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+(1-2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\frac {1}{33177600} x^{6}+\frac {1}{3251404800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}-\frac {49}{331776000} x^{6}-\frac {121}{75866112000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 174
AsymptoticDSolveValue[4*x^2*y''[x]+(1-2*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {121 x^7}{75866112000}-\frac {49 x^6}{331776000}-\frac {137 x^5}{13824000}-\frac {25 x^4}{55296}-\frac {11 x^3}{864}-\frac {3 x^2}{16}-x\right )+\sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \log (x)\right ) \]