Internal problem ID [1399]
Internal file name [OUTPUT/1400_Sunday_June_05_2022_02_15_10_AM_71217187/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: 47.
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]]
\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y=0} \] 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. \[ \left (4 x^{3}+4 x^{2}\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y = 0 \] 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) &= \frac {1}{x +1}\\ q(x) &= -\frac {5 x -1}{4 x^{2} \left (x +1\right )}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-1, 0, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y = 0 \] 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 (x +1\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right ) x^{2}+\left (1-5 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^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\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 }}4 x^{1+n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-5 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 }}4 x^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}4 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r} \\ \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 }}\left (-5 x^{1+n +r} a_{n}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-5 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 =1}{\sum }}4 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r}\right )+\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 =1}{\sum }}4 a_{n -1} \left (n +r -1\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-5 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 -1} \left (n +r -1\right ) \left (n +r -2\right )+4 a_{n} \left (n +r \right ) \left (n +r -1\right )+4 a_{n -1} \left (n +r -1\right )+a_{n}-5 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {a_{n -1} \left (4 n^{2}+8 n r +4 r^{2}-8 n -8 r -1\right )}{4 n^{2}+8 n r +4 r^{2}-4 n -4 r +1}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = -\frac {a_{n -1} \left (n^{2}-n -1\right )}{n^{2}}\tag {5} \] 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 \[ a_{1}=\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{1}=1 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}=-{\frac {1}{4}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) |
| \(a_{2}\) | \(\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}}\) | \(-{\frac {1}{4}}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {-64 r^{6}-384 r^{5}-592 r^{4}+192 r^{3}+884 r^{2}+360 r -55}{64 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {5}{36}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) |
| \(a_{2}\) | \(\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}}\) | \(-{\frac {1}{4}}\) |
| \(a_{3}\) | \(\frac {-64 r^{6}-384 r^{5}-592 r^{4}+192 r^{3}+884 r^{2}+360 r -55}{64 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2}}\) | \(\frac {5}{36}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right )}{\left (2 r +1\right )^{2} \left (3+2 r \right )^{2} \left (2 r +5\right )^{2} \left (4 r^{2}+28 r +49\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}=-{\frac {55}{576}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) |
| \(a_{2}\) | \(\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}}\) | \(-{\frac {1}{4}}\) |
| \(a_{3}\) | \(\frac {-64 r^{6}-384 r^{5}-592 r^{4}+192 r^{3}+884 r^{2}+360 r -55}{64 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2}}\) | \(\frac {5}{36}\) |
| \(a_{4}\) | \(\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right )}{\left (2 r +1\right )^{2} \left (3+2 r \right )^{2} \left (2 r +5\right )^{2} \left (4 r^{2}+28 r +49\right )}\) | \(-{\frac {55}{576}}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right ) \left (4 r^{2}+32 r +59\right )}{1024 \left (r +\frac {1}{2}\right )^{2} \left (r +\frac {9}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2} \left (r +\frac {7}{2}\right )^{2}} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {209}{2880}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) |
| \(a_{2}\) | \(\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}}\) | \(-{\frac {1}{4}}\) |
| \(a_{3}\) | \(\frac {-64 r^{6}-384 r^{5}-592 r^{4}+192 r^{3}+884 r^{2}+360 r -55}{64 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2}}\) | \(\frac {5}{36}\) |
| \(a_{4}\) | \(\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right )}{\left (2 r +1\right )^{2} \left (3+2 r \right )^{2} \left (2 r +5\right )^{2} \left (4 r^{2}+28 r +49\right )}\) | \(-{\frac {55}{576}}\) |
| \(a_{5}\) | \(-\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right ) \left (4 r^{2}+32 r +59\right )}{1024 \left (r +\frac {1}{2}\right )^{2} \left (r +\frac {9}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2} \left (r +\frac {7}{2}\right )^{2}}\) | \(\frac {209}{2880}\) |
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}\dots \right ) \\ &= \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\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 |
| \(b_{1}\) | \(\frac {-4 r^{2}+5}{\left (2 r +1\right )^{2}}\) | \(1\) | \(\frac {-8 r -20}{\left (2 r +1\right )^{3}}\) | \(-3\) |
| \(b_{2}\) | \(\frac {16 r^{4}+32 r^{3}-24 r^{2}-40 r +5}{16 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2}}\) | \(-{\frac {1}{4}}\) | \(\frac {128 r^{4}+640 r^{3}+768 r^{2}+96 r -200}{\left (3+2 r \right )^{3} \left (2 r +1\right )^{3}}\) | \(\frac {1}{4}\) |
| \(b_{3}\) | \(\frac {-64 r^{6}-384 r^{5}-592 r^{4}+192 r^{3}+884 r^{2}+360 r -55}{64 \left (r +\frac {1}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2}}\) | \(\frac {5}{36}\) | \(-\frac {4 \left (384 r^{7}+4032 r^{6}+15840 r^{5}+29616 r^{4}+26184 r^{3}+6900 r^{2}-4470 r -2615\right )}{\left (2 r +5\right )^{3} \left (3+2 r \right )^{3} \left (2 r +1\right )^{3}}\) | \(-{\frac {5}{54}}\) |
| \(b_{4}\) | \(\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right )}{\left (2 r +1\right )^{2} \left (3+2 r \right )^{2} \left (2 r +5\right )^{2} \left (4 r^{2}+28 r +49\right )}\) | \(-{\frac {55}{576}}\) | \(\frac {16384 r^{10}+311296 r^{9}+2494464 r^{8}+11059200 r^{7}+29755392 r^{6}+49735680 r^{5}+49777152 r^{4}+24943616 r^{3}-166080 r^{2}-6405440 r -2233520}{\left (2 r +1\right )^{3} \left (3+2 r \right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3}}\) | \(\frac {175}{3456}\) |
| \(b_{5}\) | \(-\frac {\left (4 r^{2}+24 r +31\right ) \left (4 r^{2}+16 r +11\right ) \left (4 r^{2}-5\right ) \left (4 r^{2}+8 r -1\right ) \left (4 r^{2}+32 r +59\right )}{1024 \left (r +\frac {1}{2}\right )^{2} \left (r +\frac {9}{2}\right )^{2} \left (\frac {3}{2}+r \right )^{2} \left (r +\frac {5}{2}\right )^{2} \left (r +\frac {7}{2}\right )^{2}}\) | \(\frac {209}{2880}\) | \(-\frac {20 \left (8192 r^{13}+249856 r^{12}+3379200 r^{11}+26781696 r^{10}+138325504 r^{9}+489143040 r^{8}+1208225280 r^{7}+2080189184 r^{6}+2424931296 r^{5}+1758393904 r^{4}+574475920 r^{3}-163041240 r^{2}-216459182 r -58834491\right )}{\left (2 r +1\right )^{3} \left (2 r +9\right )^{3} \left (3+2 r \right )^{3} \left (2 r +5\right )^{3} \left (2 r +7\right )^{3}}\) | \(-{\frac {2863}{86400}}\) |
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}\dots \\ &= \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\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 (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) + c_{2} \left (\sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\right )\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\right )\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\right )\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \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 {\left (5 x -1\right ) y}{4 x^{2} \left (x +1\right )}-\frac {y^{\prime }}{x +1} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+\frac {y^{\prime }}{x +1}-\frac {\left (5 x -1\right ) y}{4 x^{2} \left (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 {1}{x +1}, P_{3}\left (x \right )=-\frac {5 x -1}{4 x^{2} \left (x +1\right )}\right ] \\ {} & \circ & \left (x +1\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=1 \\ {} & \circ & \left (x +1\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-1 \\ {} & {} & \left (\left (x +1\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-1}}}=0 \\ {} & \circ & x =-1\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}=-1 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -1\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (4 u^{3}-8 u^{2}+4 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (4 u^{2}-8 u +4\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (6-5 u \right ) y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 4 a_{0} r^{2} u^{-1+r}+\left (4 a_{1} \left (1+r \right )^{2}-2 a_{0} \left (4 r^{2}-3\right )\right ) u^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (4 a_{k +1} \left (k +1+r \right )^{2}-2 a_{k} \left (4 k^{2}+8 k r +4 r^{2}-3\right )+a_{k -1} \left (4 \left (k -1\right )^{2}+8 \left (k -1\right ) r +4 r^{2}-5\right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 4 r^{2}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r =0 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & 4 a_{1} \left (1+r \right )^{2}-2 a_{0} \left (4 r^{2}-3\right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-8 a_{k}+4 a_{k -1}+4 a_{k +1}\right ) k^{2}+\left (-8 a_{k -1}+8 a_{k +1}\right ) k +6 a_{k}-a_{k -1}+4 a_{k +1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (-8 a_{k +1}+4 a_{k}+4 a_{k +2}\right ) \left (k +1\right )^{2}+\left (-8 a_{k}+8 a_{k +2}\right ) \left (k +1\right )+6 a_{k +1}-a_{k}+4 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}-8 k^{2} a_{k +1}-16 k a_{k +1}-5 a_{k}-2 a_{k +1}}{4 \left (k^{2}+4 k +4\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {4 k^{2} a_{k}-8 k^{2} a_{k +1}-16 k a_{k +1}-5 a_{k}-2 a_{k +1}}{4 \left (k^{2}+4 k +4\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +2}=-\frac {4 k^{2} a_{k}-8 k^{2} a_{k +1}-16 k a_{k +1}-5 a_{k}-2 a_{k +1}}{4 \left (k^{2}+4 k +4\right )}, 4 a_{1}+6 a_{0}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +1\right )^{k}, a_{k +2}=-\frac {4 k^{2} a_{k}-8 k^{2} a_{k +1}-16 k a_{k +1}-5 a_{k}-2 a_{k +1}}{4 \left (k^{2}+4 k +4\right )}, 4 a_{1}+6 a_{0}=0\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 -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 69
Order:=6; dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x^2*diff(y(x),x)+(1-5*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x -\frac {1}{4} x^{2}+\frac {5}{36} x^{3}-\frac {55}{576} x^{4}+\frac {209}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x +\frac {1}{4} x^{2}-\frac {5}{54} x^{3}+\frac {175}{3456} x^{4}-\frac {2863}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 124
AsymptoticDSolveValue[4*x^2*(1+x)*y''[x]+4*x^2*y'[x]+(1-5*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {2863 x^5}{86400}+\frac {175 x^4}{3456}-\frac {5 x^3}{54}+\frac {x^2}{4}-3 x\right )+\sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right ) \log (x)\right ) \]