Internal problem ID [4704]
Internal file name [OUTPUT/4197_Sunday_June_05_2022_12_39_59_PM_6718125/index.tex]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 10.
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]]
\[ \boxed {\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y=0} \] 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. \[ \left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }+\left (-6 x^{2}+7 x -1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] 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 {6 x^{2}-7 x +1}{2 x \left (2 x^{2}-7 x -1\right )}\\ q(x) &= \frac {6 x -1}{2 x \left (2 x^{2}-7 x -1\right )}\\ \end {align*}
| \(p(x)=-\frac {6 x^{2}-7 x +1}{2 x \left (2 x^{2}-7 x -1\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = \frac {7}{4}-\frac {\sqrt {57}}{4}\) | \(\text {``regular''}\) |
| \(x = \frac {7}{4}+\frac {\sqrt {57}}{4}\) | \(\text {``regular''}\) |
| \(q(x)=\frac {6 x -1}{2 x \left (2 x^{2}-7 x -1\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = \frac {7}{4}-\frac {\sqrt {57}}{4}\) | \(\text {``regular''}\) |
| \(x = \frac {7}{4}+\frac {\sqrt {57}}{4}\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \(\left [0, \frac {7}{4}-\frac {\sqrt {57}}{4}, \frac {7}{4}+\frac {\sqrt {57}}{4}, \infty \right ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 y^{\prime \prime } x \left (2 x^{2}-7 x -1\right )+\left (-6 x^{2}+7 x -1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] 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 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right ) x \left (2 x^{2}-7 x -1\right )+\left (-6 x^{2}+7 x -1\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (6 x -1\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^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-14 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-6 x^{1+n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}7 x^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-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 }}4 x^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}4 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right ) x^{n +r -1} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-14 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-14 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-6 x^{1+n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-6 a_{n -2} \left (n +r -2\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}7 x^{n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}7 a_{n -1} \left (n +r -1\right ) x^{n +r -1} \\ \moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n} &= \moverset {\infty }{\munderset {n =2}{\sum }}6 a_{n -2} x^{n +r -1} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} x^{n +r}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r -1}\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 \(n +r -1\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =2}{\sum }}4 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right ) x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-14 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-6 a_{n -2} \left (n +r -2\right ) x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}7 a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}6 a_{n -2} x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ -2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )-\left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ -2 x^{-1+r} a_{0} r \left (-1+r \right )-r a_{0} x^{-1+r} = 0 \] Or \[ \left (-2 x^{-1+r} r \left (-1+r \right )-r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (-2 r^{2}+r \right ) x^{-1+r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ -2 r^{2}+r = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {1}{2}}\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (-2 r^{2}+r \right ) x^{-1+r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {1}{2}}, 0\right ]\).
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 {1}{2}}\\ 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 \(a_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ a_{1} = \frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 4 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right )-14 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right )-2 a_{n} \left (n +r \right ) \left (n +r -1\right )-6 a_{n -2} \left (n +r -2\right )+7 a_{n -1} \left (n +r -1\right )-a_{n} \left (n +r \right )+6 a_{n -2}-a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {4 n^{2} a_{n -2}-14 n^{2} a_{n -1}+8 n r a_{n -2}-28 n r a_{n -1}+4 r^{2} a_{n -2}-14 r^{2} a_{n -1}-26 n a_{n -2}+49 n a_{n -1}-26 r a_{n -2}+49 r a_{n -1}+42 a_{n -2}-36 a_{n -1}}{2 n^{2}+4 n r +2 r^{2}-n -r}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = \frac {\left (4 a_{n -2}-14 a_{n -1}\right ) n^{2}+\left (-22 a_{n -2}+35 a_{n -1}\right ) n +30 a_{n -2}-15 a_{n -1}}{2 n^{2}+n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {1}{2}}\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(2\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(2\) |
| \(a_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(2\) |
| \(a_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(a_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(2\) |
| \(a_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(a_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
| \(a_{4}\) | \(\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {-78540 r^{4}-157080 r^{3}+548853 r^{2}-225339 r}{4 r^{4}+68 r^{3}+431 r^{2}+1207 r +1260} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(2\) |
| \(a_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(a_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
| \(a_{4}\) | \(\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420}\) | \(0\) |
| \(a_{5}\) | \(\frac {-78540 r^{4}-157080 r^{3}+548853 r^{2}-225339 r}{4 r^{4}+68 r^{3}+431 r^{2}+1207 r +1260}\) | \(0\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= \sqrt {x} \left (a_{0}+a_{1} x +a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}+a_{6} x^{6}\dots \right ) \\ &= \sqrt {x}\, \left (1+2 x +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 \(b_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ b_{1} = \frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 4 b_{n -2} \left (n +r -2\right ) \left (n -3+r \right )-14 b_{n -1} \left (n +r -1\right ) \left (n +r -2\right )-2 b_{n} \left (n +r \right ) \left (n +r -1\right )-6 b_{n -2} \left (n +r -2\right )+7 b_{n -1} \left (n +r -1\right )-\left (n +r \right ) b_{n}+6 b_{n -2}-b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = \frac {4 n^{2} b_{n -2}-14 n^{2} b_{n -1}+8 n r b_{n -2}-28 n r b_{n -1}+4 r^{2} b_{n -2}-14 r^{2} b_{n -1}-26 n b_{n -2}+49 n b_{n -1}-26 r b_{n -2}+49 r b_{n -1}+42 b_{n -2}-36 b_{n -1}}{2 n^{2}+4 n r +2 r^{2}-n -r}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = \frac {\left (4 b_{n -2}-14 b_{n -1}\right ) n^{2}+\left (-26 b_{n -2}+49 b_{n -1}\right ) n +42 b_{n -2}-36 b_{n -1}}{2 n^{2}-n}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = 0\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(-1\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6} \] Which for the root \(r = 0\) becomes \[ b_{2}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(-1\) |
| \(b_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90} \] Which for the root \(r = 0\) becomes \[ b_{3}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(-1\) |
| \(b_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(b_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420} \] Which for the root \(r = 0\) becomes \[ b_{4}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(-1\) |
| \(b_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(b_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
| \(b_{4}\) | \(\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {-78540 r^{4}-157080 r^{3}+548853 r^{2}-225339 r}{4 r^{4}+68 r^{3}+431 r^{2}+1207 r +1260} \] Which for the root \(r = 0\) becomes \[ b_{5}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {-14 r^{2}+21 r -1}{2 r^{2}+3 r +1}\) | \(-1\) |
| \(b_{2}\) | \(\frac {204 r^{4}-204 r^{3}-231 r^{2}+141 r}{4 r^{4}+20 r^{3}+35 r^{2}+25 r +6}\) | \(0\) |
| \(b_{3}\) | \(\frac {-1484 r^{4}+4577 r^{2}-2103 r}{4 r^{4}+36 r^{3}+119 r^{2}+171 r +90}\) | \(0\) |
| \(b_{4}\) | \(\frac {10796 r^{4}+10796 r^{3}-54371 r^{2}+23137 r}{4 r^{4}+52 r^{3}+251 r^{2}+533 r +420}\) | \(0\) |
| \(b_{5}\) | \(\frac {-78540 r^{4}-157080 r^{3}+548853 r^{2}-225339 r}{4 r^{4}+68 r^{3}+431 r^{2}+1207 r +1260}\) | \(0\) |
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}\dots \\ &= 1-x +O\left (x^{6}\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+2 x +O\left (x^{6}\right )\right ) + c_{2} \left (1-x +O\left (x^{6}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (1+2 x +O\left (x^{6}\right )\right )+c_{2} \left (1-x +O\left (x^{6}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \left (1+2 x +O\left (x^{6}\right )\right )+c_{2} \left (1-x +O\left (x^{6}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1+2 x +O\left (x^{6}\right )\right )+c_{2} \left (1-x +O\left (x^{6}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \left (\frac {d}{d x}y^{\prime }\right ) x \left (2 x^{2}-7 x -1\right )+\left (-6 x^{2}+7 x -1\right ) y^{\prime }+\left (6 x -1\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 (6 x -1\right ) y}{2 x \left (2 x^{2}-7 x -1\right )}+\frac {\left (6 x^{2}-7 x +1\right ) y^{\prime }}{2 x \left (2 x^{2}-7 x -1\right )} \\ \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 (6 x^{2}-7 x +1\right ) y^{\prime }}{2 x \left (2 x^{2}-7 x -1\right )}+\frac {\left (6 x -1\right ) y}{2 x \left (2 x^{2}-7 x -1\right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=-\frac {6 x^{2}-7 x +1}{2 x \left (2 x^{2}-7 x -1\right )}, P_{3}\left (x \right )=\frac {6 x -1}{2 x \left (2 x^{2}-7 x -1\right )}\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}}}=0 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 \left (\frac {d}{d x}y^{\prime }\right ) x \left (2 x^{2}-7 x -1\right )+\left (-6 x^{2}+7 x -1\right ) y^{\prime }+\left (6 x -1\right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..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^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\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 -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} r \left (-1+2 r \right ) x^{-1+r}+\left (-a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (14 r^{2}-21 r +1\right )\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a_{k +1} \left (k +1+r \right ) \left (2 k +1+2 r \right )-a_{k} \left (14 k^{2}+28 k r +14 r^{2}-21 k -21 r +1\right )+2 a_{k -1} \left (k -2+r \right ) \left (2 k -5+2 r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -r \left (-1+2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {1}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & -a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (14 r^{2}-21 r +1\right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-14 a_{k}+4 a_{k -1}-2 a_{k +1}\right ) k^{2}+\left (\left (-28 a_{k}+8 a_{k -1}-4 a_{k +1}\right ) r +21 a_{k}-18 a_{k -1}-3 a_{k +1}\right ) k +\left (-14 a_{k}+4 a_{k -1}-2 a_{k +1}\right ) r^{2}+\left (21 a_{k}-18 a_{k -1}-3 a_{k +1}\right ) r -a_{k}+20 a_{k -1}-a_{k +1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (-14 a_{k +1}+4 a_{k}-2 a_{k +2}\right ) \left (k +1\right )^{2}+\left (\left (-28 a_{k +1}+8 a_{k}-4 a_{k +2}\right ) r +21 a_{k +1}-18 a_{k}-3 a_{k +2}\right ) \left (k +1\right )+\left (-14 a_{k +1}+4 a_{k}-2 a_{k +2}\right ) r^{2}+\left (21 a_{k +1}-18 a_{k}-3 a_{k +2}\right ) r -a_{k +1}+20 a_{k}-a_{k +2}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}+8 k r a_{k}-28 k r a_{k +1}+4 r^{2} a_{k}-14 r^{2} a_{k +1}-10 k a_{k}-7 k a_{k +1}-10 r a_{k}-7 r a_{k +1}+6 a_{k}+6 a_{k +1}}{2 k^{2}+4 k r +2 r^{2}+7 k +7 r +6} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}-10 k a_{k}-7 k a_{k +1}+6 a_{k}+6 a_{k +1}}{2 k^{2}+7 k +6} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}-10 k a_{k}-7 k a_{k +1}+6 a_{k}+6 a_{k +1}}{2 k^{2}+7 k +6}, -a_{1}-a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}-6 k a_{k}-21 k a_{k +1}+2 a_{k}-a_{k +1}}{2 k^{2}+9 k +10} \\ \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 +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}-6 k a_{k}-21 k a_{k +1}+2 a_{k}-a_{k +1}}{2 k^{2}+9 k +10}, -3 a_{1}+6 a_{0}=0\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 {1}{2}}\right ), a_{k +2}=\frac {4 k^{2} a_{k}-14 k^{2} a_{k +1}-10 k a_{k}-7 k a_{k +1}+6 a_{k}+6 a_{k +1}}{2 k^{2}+7 k +6}, -a_{1}-a_{0}=0, b_{k +2}=\frac {4 k^{2} b_{k}-14 k^{2} b_{k +1}-6 k b_{k}-21 k b_{k +1}+2 b_{k}-b_{k +1}}{2 k^{2}+9 k +10}, -3 b_{1}+6 b_{0}=0\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) Reducible group (found another exponential solution) <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 28
Order:=6; dsolve((4*x^3-14*x^2-2*x)*diff(y(x),x$2)-(6*x^2-7*x+1)*diff(y(x),x)+(6*x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+2 x +\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 25
AsymptoticDSolveValue[(4*x^3-14*x^2-2*x)*y''[x]-(6*x^2-7*x+1)*y'[x]+(6*x-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} (2 x+1)+c_2 (1-x) \]