Internal problem ID [1360]
Internal file name [OUTPUT/1361_Sunday_June_05_2022_02_13_06_AM_50960082/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF
FROBENIUS II. Exercises 7.6. Page 374
Problem number: 8.
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]]
\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0} \] 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. \[ \left (2 x^{3}+x^{2}\right ) y^{\prime \prime }+\left (3 x^{3}+14 x^{2}+5 x \right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] 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 {3 x^{2}+14 x +5}{x \left (1+2 x \right )}\\ q(x) &= \frac {12 x^{2}+18 x +4}{x^{2} \left (1+2 x \right )}\\ \end {align*}
| \(p(x)=\frac {3 x^{2}+14 x +5}{x \left (1+2 x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = -{\frac {1}{2}}\) | \(\text {``regular''}\) |
| \(q(x)=\frac {12 x^{2}+18 x +4}{x^{2} \left (1+2 x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = -{\frac {1}{2}}\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \(\left [0, -{\frac {1}{2}}\right ]\)
Irregular singular points : \([\infty ]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ x^{2} \left (1+2 x \right ) y^{\prime \prime }+\left (3 x^{3}+14 x^{2}+5 x \right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] Let the solution be represented as Frobenius power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} \] Then \begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} \\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2} \\ \end{align*} Substituting the above back into the ode gives \begin{equation} \tag{1} x^{2} \left (1+2 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+\left (3 x^{3}+14 x^{2}+5 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (12 x^{2}+18 x +4\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^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 x^{n +r +2} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}14 x^{1+n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}5 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}12 x^{n +r +2} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}18 x^{1+n +r} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 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 }}2 x^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}3 x^{n +r +2} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}3 a_{n -2} \left (n +r -2\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}14 x^{1+n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}14 a_{n -1} \left (n +r -1\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}12 x^{n +r +2} a_{n} &= \moverset {\infty }{\munderset {n =2}{\sum }}12 a_{n -2} x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}18 x^{1+n +r} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}18 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 =1}{\sum }}2 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}3 a_{n -2} \left (n +r -2\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}14 a_{n -1} \left (n +r -1\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}5 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}12 a_{n -2} x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}18 a_{n -1} x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+5 x^{n +r} a_{n} \left (n +r \right )+4 a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )+5 x^{r} a_{0} r +4 a_{0} x^{r} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )+5 x^{r} r +4 x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (2+r \right )^{2} x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ \left (2+r \right )^{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= -2\\ r_2 &= -2 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (2+r \right )^{2} x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([-2, -2]\).
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, \(r = -2\), Eqs (1A,1B) become \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n -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 -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 \(a_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ a_{1} = -2 \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 2 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+a_{n} \left (n +r \right ) \left (n +r -1\right )+3 a_{n -2} \left (n +r -2\right )+14 a_{n -1} \left (n +r -1\right )+5 a_{n} \left (n +r \right )+12 a_{n -2}+18 a_{n -1}+4 a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {2 n a_{n -1}+2 r a_{n -1}+3 a_{n -2}+4 a_{n -1}}{n +r +2}\tag {4} \] Which for the root \(r = -2\) becomes \[ a_{n} = \frac {-2 n a_{n -1}-3 a_{n -2}}{n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = -2\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {4 r +13}{4+r} \] Which for the root \(r = -2\) becomes \[ a_{2}={\frac {5}{2}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = -2\) becomes \[ a_{3}=-3 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
| \(a_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = -2\) becomes \[ a_{4}={\frac {33}{8}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
| \(a_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) |
| \(a_{4}\) | \(\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {33}{8}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {-32 r^{4}-608 r^{3}-4198 r^{2}-12418 r -13170}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )} \] Which for the root \(r = -2\) becomes \[ a_{5}=-{\frac {129}{20}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
| \(a_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) |
| \(a_{4}\) | \(\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {33}{8}\) |
| \(a_{5}\) | \(\frac {-32 r^{4}-608 r^{3}-4198 r^{2}-12418 r -13170}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(-{\frac {129}{20}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {64 r^{5}+1680 r^{4}+17176 r^{3}+85221 r^{2}+204304 r +188103}{\left (8+r \right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )} \] Which for the root \(r = -2\) becomes \[ a_{6}={\frac {867}{80}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
| \(a_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) |
| \(a_{4}\) | \(\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {33}{8}\) |
| \(a_{5}\) | \(\frac {-32 r^{4}-608 r^{3}-4198 r^{2}-12418 r -13170}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(-{\frac {129}{20}}\) |
| \(a_{6}\) | \(\frac {64 r^{5}+1680 r^{4}+17176 r^{3}+85221 r^{2}+204304 r +188103}{\left (8+r \right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(\frac {867}{80}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {-128 r^{6}-4416 r^{5}-62000 r^{4}-452424 r^{3}-1804580 r^{2}-3716136 r -3069774}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right ) \left (8+r \right ) \left (9+r \right )} \] Which for the root \(r = -2\) becomes \[ a_{7}=-{\frac {1059}{56}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-2\) | \(-2\) |
| \(a_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) |
| \(a_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) |
| \(a_{4}\) | \(\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {33}{8}\) |
| \(a_{5}\) | \(\frac {-32 r^{4}-608 r^{3}-4198 r^{2}-12418 r -13170}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(-{\frac {129}{20}}\) |
| \(a_{6}\) | \(\frac {64 r^{5}+1680 r^{4}+17176 r^{3}+85221 r^{2}+204304 r +188103}{\left (8+r \right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(\frac {867}{80}\) |
| \(a_{7}\) | \(\frac {-128 r^{6}-4416 r^{5}-62000 r^{4}-452424 r^{3}-1804580 r^{2}-3716136 r -3069774}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right ) \left (8+r \right ) \left (9+r \right )}\) | \(-{\frac {1059}{56}}\) |
Using the above table, then the first solution \(y_{1}\left (x \right )\) is \begin{align*} y_{1}\left (x \right )&= \frac {1}{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 ) \\ &= \frac {1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )}{x^{2}} \\ \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 \(r = -2\). The above table for \(a_{n ,r}\) is used for this purpose. Computing the derivatives gives the following table
| \(n\) | \(b_{n ,r}\) | \(a_{n}\) | \(b_{n ,r} = \frac {d}{d r}a_{n ,r}\) | \(b_{n}\left (r =-2\right )\) |
| \(b_{0}\) | \(1\) | \(1\) | N/A since \(b_{n}\) starts from 1 | N/A |
| \(b_{1}\) | \(-2\) | \(-2\) | \(0\) | \(0\) |
| \(b_{2}\) | \(\frac {4 r +13}{4+r}\) | \(\frac {5}{2}\) | \(\frac {3}{\left (4+r \right )^{2}}\) | \(\frac {3}{4}\) |
| \(b_{3}\) | \(\frac {-8 r^{2}-60 r -106}{\left (4+r \right ) \left (5+r \right )}\) | \(-3\) | \(\frac {-12 r^{2}-108 r -246}{\left (4+r \right )^{2} \left (5+r \right )^{2}}\) | \(-{\frac {13}{6}}\) |
| \(b_{4}\) | \(\frac {16 r^{3}+204 r^{2}+833 r +1077}{\left (6+r \right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {33}{8}\) | \(\frac {36 r^{4}+702 r^{3}+5130 r^{2}+16650 r +20262}{\left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2}}\) | \(\frac {407}{96}\) |
| \(b_{5}\) | \(\frac {-32 r^{4}-608 r^{3}-4198 r^{2}-12418 r -13170}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(-{\frac {129}{20}}\) | \(-\frac {6 \left (16 r^{6}+510 r^{5}+6745 r^{4}+47376 r^{3}+186407 r^{2}+389630 r +338110\right )}{\left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2} \left (7+r \right )^{2}}\) | \(-{\frac {9047}{1200}}\) |
| \(b_{6}\) | \(\frac {64 r^{5}+1680 r^{4}+17176 r^{3}+85221 r^{2}+204304 r +188103}{\left (8+r \right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(\frac {867}{80}\) | \(\frac {240 r^{8}+11088 r^{7}+222897 r^{6}+2546388 r^{5}+18081150 r^{4}+81718488 r^{3}+229582809 r^{2}+366623820 r +254838648}{\left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2} \left (7+r \right )^{2} \left (8+r \right )^{2}}\) | \(\frac {63851}{4800}\) |
| \(b_{7}\) | \(\frac {-128 r^{6}-4416 r^{5}-62000 r^{4}-452424 r^{3}-1804580 r^{2}-3716136 r -3069774}{\left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right ) \left (8+r \right ) \left (9+r \right )}\) | \(-{\frac {1059}{56}}\) | \(-\frac {6 \left (96 r^{10}+6000 r^{9}+167748 r^{8}+2762496 r^{7}+29673842 r^{6}+217236534 r^{5}+1097661191 r^{4}+3780003348 r^{3}+8490832241 r^{2}+11234790708 r +6650399016\right )}{\left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2} \left (7+r \right )^{2} \left (8+r \right )^{2} \left (9+r \right )^{2}}\) | \(-{\frac {559033}{23520}}\) |
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 \\ &= \frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}} \\ \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 ) \\ &= \frac {c_{1} \left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right )}{x^{2}} + c_{2} \left (\frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= \frac {c_{1} \left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} \left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} \left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (1+2 x \right ) y^{\prime \prime }+\left (3 x^{3}+14 x^{2}+5 x \right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {2 \left (6 x^{2}+9 x +2\right ) y}{x^{2} \left (1+2 x \right )}-\frac {\left (3 x^{2}+14 x +5\right ) y^{\prime }}{x \left (1+2 x \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} \\ {} & {} & y^{\prime \prime }+\frac {\left (3 x^{2}+14 x +5\right ) y^{\prime }}{x \left (1+2 x \right )}+\frac {2 \left (6 x^{2}+9 x +2\right ) y}{x^{2} \left (1+2 x \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 {3 x^{2}+14 x +5}{x \left (1+2 x \right )}, P_{3}\left (x \right )=\frac {2 \left (6 x^{2}+9 x +2\right )}{x^{2} \left (1+2 x \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}}}=5 \\ {} & \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}}}=4 \\ {} & \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} \\ {} & {} & x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\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..2 \\ {} & {} & 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..3 \\ {} & {} & 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 y^{\prime \prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..3 \\ {} & {} & x^{m}\cdot y^{\prime \prime }=\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 y^{\prime \prime }=\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} \left (2+r \right )^{2} x^{r}+\left (a_{1} \left (3+r \right )^{2}+2 a_{0} \left (3+r \right )^{2}\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k +r +2\right )^{2}+2 a_{k -1} \left (k +r +2\right )^{2}+3 a_{k -2} \left (k +r +2\right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (2+r \right )^{2}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r =-2 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (3+r \right )^{2}+2 a_{0} \left (3+r \right )^{2}=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-2 a_{0} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (\left (2 k +2 r +4\right ) a_{k -1}+a_{k} \left (k +r +2\right )+3 a_{k -2}\right ) \left (k +r +2\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (2 k +8+2 r \right ) a_{k +1}+a_{k +2} \left (k +r +4\right )+3 a_{k}\right ) \left (k +r +4\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {2 k a_{k +1}+2 r a_{k +1}+3 a_{k}+8 a_{k +1}}{k +r +4} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-2 \\ {} & {} & a_{k +2}=-\frac {2 k a_{k +1}+3 a_{k}+4 a_{k +1}}{k +2} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-2 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -2}, a_{k +2}=-\frac {2 k a_{k +1}+3 a_{k}+4 a_{k +1}}{k +2}, a_{1}=-2 a_{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) 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 -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c = 0 <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 79
Order:=8; dsolve(x^2*(1+2*x)*diff(y(x),x$2)+x*(5+14*x+3*x^2)*diff(y(x),x)+(4+18*x+12*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {5}{2} x^{2}-3 x^{3}+\frac {33}{8} x^{4}-\frac {129}{20} x^{5}+\frac {867}{80} x^{6}-\frac {1059}{56} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{4} x^{2}-\frac {13}{6} x^{3}+\frac {407}{96} x^{4}-\frac {9047}{1200} x^{5}+\frac {63851}{4800} x^{6}-\frac {559033}{23520} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 157
AsymptoticDSolveValue[x^2*(1+2*x)*y''[x]+x*(5+14*x+3*x^2)*y'[x]+(4+18*x+12*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {1059 x^7}{56}+\frac {867 x^6}{80}-\frac {129 x^5}{20}+\frac {33 x^4}{8}-3 x^3+\frac {5 x^2}{2}-2 x+1\right )}{x^2}+c_2 \left (\frac {-\frac {559033 x^7}{23520}+\frac {63851 x^6}{4800}-\frac {9047 x^5}{1200}+\frac {407 x^4}{96}-\frac {13 x^3}{6}+\frac {3 x^2}{4}}{x^2}+\frac {\left (-\frac {1059 x^7}{56}+\frac {867 x^6}{80}-\frac {129 x^5}{20}+\frac {33 x^4}{8}-3 x^3+\frac {5 x^2}{2}-2 x+1\right ) \log (x)}{x^2}\right ) \]