Internal problem ID [6932]
Internal file name [OUTPUT/6175_Friday_August_12_2022_11_04_23_PM_19162768/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 16.
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]]
\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) y=0} \] 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. \[ 2 x^{2} y^{\prime \prime }+\left (4 x^{2}-x \right ) y^{\prime }+\left (6 x -2\right ) y = 0 \] 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 {4 x -1}{2 x}\\ q(x) &= \frac {3 x -1}{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 \[ 2 x^{2} y^{\prime \prime }+\left (4 x^{2}-x \right ) y^{\prime }+\left (6 x -2\right ) y = 0 \] 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} 2 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 (4 x^{2}-x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (6 x -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 }}2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{1+n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 a_{n} x^{n +r}\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 }}4 x^{1+n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}4 a_{n -1} \left (n +r -1\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} x^{n +r} \\ \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 }}2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}4 a_{n -1} \left (n +r -1\right ) x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )-x^{n +r} a_{n} \left (n +r \right )-2 a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ 2 x^{r} a_{0} r \left (-1+r \right )-x^{r} a_{0} r -2 a_{0} x^{r} = 0 \] Or \[ \left (2 x^{r} r \left (-1+r \right )-x^{r} r -2 x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (2 r^{2}-3 r -2\right ) x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 2 r^{2}-3 r -2 = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 2\\ r_2 &= -{\frac {1}{2}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (2 r^{2}-3 r -2\right ) x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [2, -{\frac {1}{2}}\right ]\).
Since \(r_1 - r_2 = {\frac {5}{2}}\) 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 +2}\\ y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -\frac {1}{2}} \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} 2 a_{n} \left (n +r \right ) \left (n +r -1\right )+4 a_{n -1} \left (n +r -1\right )-a_{n} \left (n +r \right )+6 a_{n -1}-2 a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {2 a_{n -1}}{n +r -2}\tag {4} \] Which for the root \(r = 2\) becomes \[ a_{n} = -\frac {2 a_{n -1}}{n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 2\) 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 \[ a_{1}=-\frac {2}{-1+r} \] Which for the root \(r = 2\) becomes \[ a_{1}=-2 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {4}{\left (-1+r \right ) r} \] Which for the root \(r = 2\) becomes \[ a_{2}=2 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {8}{r^{3}-r} \] Which for the root \(r = 2\) becomes \[ a_{3}=-{\frac {4}{3}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
| \(a_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {4}{3}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{4}={\frac {2}{3}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
| \(a_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {4}{3}}\) |
| \(a_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {2}{3}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r} \] Which for the root \(r = 2\) becomes \[ a_{5}=-{\frac {4}{15}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
| \(a_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {4}{3}}\) |
| \(a_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {2}{3}\) |
| \(a_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {4}{15}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{6}={\frac {4}{45}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
| \(a_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {4}{3}}\) |
| \(a_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {2}{3}\) |
| \(a_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {4}{15}}\) |
| \(a_{6}\) | \(\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )}\) | \(\frac {4}{45}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=-\frac {128}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{7}=-{\frac {8}{315}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {2}{-1+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(2\) |
| \(a_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {4}{3}}\) |
| \(a_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {2}{3}\) |
| \(a_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {4}{15}}\) |
| \(a_{6}\) | \(\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )}\) | \(\frac {4}{45}\) |
| \(a_{7}\) | \(-\frac {128}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right ) \left (5+r \right )}\) | \(-{\frac {8}{315}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{2} \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^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+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} 2 b_{n} \left (n +r \right ) \left (n +r -1\right )+4 b_{n -1} \left (n +r -1\right )-b_{n} \left (n +r \right )+6 b_{n -1}-2 b_{n} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {2 b_{n -1}}{n +r -2}\tag {4} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{n} = -\frac {4 b_{n -1}}{2 n -5}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = -{\frac {1}{2}}\) 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 \[ b_{1}=-\frac {2}{-1+r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{1}={\frac {4}{3}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {4}{\left (-1+r \right ) r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{2}={\frac {16}{3}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {8}{r^{3}-r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{3}=-{\frac {64}{3}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
| \(b_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {64}{3}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{4}={\frac {256}{9}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
| \(b_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {64}{3}}\) |
| \(b_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {256}{9}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{5}=-{\frac {1024}{45}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
| \(b_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {64}{3}}\) |
| \(b_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {256}{9}\) |
| \(b_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {1024}{45}}\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{6}={\frac {4096}{315}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
| \(b_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {64}{3}}\) |
| \(b_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {256}{9}\) |
| \(b_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {1024}{45}}\) |
| \(b_{6}\) | \(\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )}\) | \(\frac {4096}{315}\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=-\frac {128}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{7}=-{\frac {16384}{2835}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {2}{-1+r}\) | \(\frac {4}{3}\) |
| \(b_{2}\) | \(\frac {4}{\left (-1+r \right ) r}\) | \(\frac {16}{3}\) |
| \(b_{3}\) | \(-\frac {8}{r^{3}-r}\) | \(-{\frac {64}{3}}\) |
| \(b_{4}\) | \(\frac {16}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(\frac {256}{9}\) |
| \(b_{5}\) | \(-\frac {32}{r^{5}+5 r^{4}+5 r^{3}-5 r^{2}-6 r}\) | \(-{\frac {1024}{45}}\) |
| \(b_{6}\) | \(\frac {64}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right )}\) | \(\frac {4096}{315}\) |
| \(b_{7}\) | \(-\frac {128}{r \left (r^{4}+5 r^{3}+5 r^{2}-5 r -6\right ) \left (4+r \right ) \left (5+r \right )}\) | \(-{\frac {16384}{2835}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= x^{2} \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}+b_{7} x^{7}+b_{8} x^{8}\dots \right ) \\ &= \frac {1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )}{\sqrt {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} x^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right ) + \frac {c_{2} \left (1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )\right )}{\sqrt {x}} \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )\right )}{\sqrt {x}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )\right )}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (4 x^{2}-x \right ) y^{\prime }+\left (6 x -2\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 (3 x -1\right ) y}{x^{2}}-\frac {\left (4 x -1\right ) y^{\prime }}{2 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} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (4 x -1\right ) y^{\prime }}{2 x}+\frac {\left (3 x -1\right ) y}{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 {4 x -1}{2 x}, P_{3}\left (x \right )=\frac {3 x -1}{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 {1}{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}}}=-1 \\ {} & \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} \left (\frac {d}{d x}y^{\prime }\right )+x \left (4 x -1\right ) y^{\prime }+\left (6 x -2\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^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) 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 ) \left (-2+r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (2 k +2 r +1\right ) \left (k +r -2\right )+2 a_{k -1} \left (2 k +2 r +1\right )\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 ) \left (-2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{2, -\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 (a_{k} \left (k +r -2\right )+2 a_{k -1}\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 2 \left (k +\frac {3}{2}+r \right ) \left (a_{k +1} \left (k +r -1\right )+2 a_{k}\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{k +r -1} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{k +1} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +2}, a_{k +1}=-\frac {2 a_{k}}{k +1}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {1}{2} \\ {} & {} & a_{k +1}=-\frac {2 a_{k}}{k -\frac {3}{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}}{k -\frac {3}{2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +2}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k -\frac {1}{2}}\right ), a_{k +1}=-\frac {2 a_{k}}{k +1}, b_{k +1}=-\frac {2 b_{k}}{k -\frac {3}{2}}\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 Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Kummer successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form is not straightforward to achieve - returning special function solution free of uncomputed integrals <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 55
Order:=8; dsolve(2*x^2*diff(y(x),x$2)+x*(4*x-1)*diff(y(x),x)+2*(3*x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {4}{3} x +\frac {16}{3} x^{2}-\frac {64}{3} x^{3}+\frac {256}{9} x^{4}-\frac {1024}{45} x^{5}+\frac {4096}{315} x^{6}-\frac {16384}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} x^{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\frac {4}{45} x^{6}-\frac {8}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 112
AsymptoticDSolveValue[2*x^2*y''[x]+x*(4*x-1)*y'[x]+2*(3*x-1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {8 x^7}{315}+\frac {4 x^6}{45}-\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1\right ) x^2+\frac {c_2 \left (-\frac {16384 x^7}{2835}+\frac {4096 x^6}{315}-\frac {1024 x^5}{45}+\frac {256 x^4}{9}-\frac {64 x^3}{3}+\frac {16 x^2}{3}+\frac {4 x}{3}+1\right )}{\sqrt {x}} \]