Internal problem ID [2397]
Internal file name [OUTPUT/2397_Tuesday_February_27_2024_08_36_52_AM_49705858/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 41, page 195
Problem number: 22.
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]]
\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }-y \left (x +1\right )=0} \] 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. \[ 2 x^{2} y^{\prime \prime }+\left (x^{3}+x \right ) y^{\prime }+\left (-x -1\right ) y = 0 \] 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 {x^{2}+1}{2 x}\\ q(x) &= -\frac {x +1}{2 x^{2}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0, \infty , -\infty ]\)
Irregular singular points : \([\infty ]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 x^{2} y^{\prime \prime }+\left (x^{3}+x \right ) y^{\prime }+\left (-x -1\right ) y = 0 \] 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 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 (x^{3}+x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (-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 }}2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r +2} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-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 \(n +r\) 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 }}x^{n +r +2} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}a_{n -2} \left (n +r -2\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{1+n +r} a_{n}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r}\right ) \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}a_{n -2} \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 )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+x^{n +r} a_{n} \left (n +r \right )-a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ 2 x^{r} a_{0} r \left (-1+r \right )+x^{r} a_{0} r -a_{0} x^{r} = 0 \] Or \[ \left (2 x^{r} r \left (-1+r \right )+x^{r} r -x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (2 r^{2}-r -1\right ) x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 2 r^{2}-r -1 = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 1\\ r_2 &= -{\frac {1}{2}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (2 r^{2}-r -1\right ) x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [1, -{\frac {1}{2}}\right ]\).
Since \(r_1 - r_2 = {\frac {3}{2}}\) 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^{1+n}\\ 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 \(a_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ a_{1} = \frac {1}{r \left (2 r +3\right )} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 2 a_{n} \left (n +r \right ) \left (n +r -1\right )+a_{n -2} \left (n +r -2\right )+a_{n} \left (n +r \right )-a_{n -1}-a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {n a_{n -2}+r a_{n -2}-2 a_{n -2}-a_{n -1}}{2 n^{2}+4 n r +2 r^{2}-n -r -1}\tag {4} \] Which for the root \(r = 1\) becomes \[ a_{n} = \frac {-n a_{n -2}+a_{n -2}+a_{n -1}}{n \left (2 n +3\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 1\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(\frac {1}{5}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r} \] Which for the root \(r = 1\) becomes \[ a_{2}=-{\frac {2}{35}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {2}{35}}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r} \] Which for the root \(r = 1\) becomes \[ a_{3}=-{\frac {16}{945}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {2}{35}}\) |
| \(a_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(-{\frac {16}{945}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )} \] Which for the root \(r = 1\) becomes \[ a_{4}={\frac {73}{20790}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {2}{35}}\) |
| \(a_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(-{\frac {16}{945}}\) |
| \(a_{4}\) | \(\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )}\) | \(\frac {73}{20790}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {12 r^{5}+132 r^{4}+549 r^{3}+1057 r^{2}+920 r +292}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right ) \left (2 r^{2}+19 r +44\right )} \] Which for the root \(r = 1\) becomes \[ a_{5}={\frac {1481}{1351350}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(\frac {1}{5}\) |
| \(a_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {2}{35}}\) |
| \(a_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(-{\frac {16}{945}}\) |
| \(a_{4}\) | \(\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )}\) | \(\frac {73}{20790}\) |
| \(a_{5}\) | \(\frac {12 r^{5}+132 r^{4}+549 r^{3}+1057 r^{2}+920 r +292}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right ) \left (2 r^{2}+19 r +44\right )}\) | \(\frac {1481}{1351350}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= 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 ) \\ &= x \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+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 \[ b_{1} = \frac {1}{r \left (2 r +3\right )} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 2 b_{n} \left (n +r \right ) \left (n +r -1\right )+b_{n -2} \left (n +r -2\right )+b_{n} \left (n +r \right )-b_{n -1}-b_{n} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {n b_{n -2}+r b_{n -2}-2 b_{n -2}-b_{n -1}}{2 n^{2}+4 n r +2 r^{2}-n -r -1}\tag {4} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{n} = \frac {-2 n b_{n -2}+5 b_{n -2}+2 b_{n -1}}{4 n^{2}-6 n}\tag {5} \] 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.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(-1\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{2}=-{\frac {1}{4}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(-1\) |
| \(b_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {1}{4}}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{3}={\frac {1}{36}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(-1\) |
| \(b_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {1}{4}}\) |
| \(b_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(\frac {1}{36}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{4}={\frac {29}{1440}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(-1\) |
| \(b_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {1}{4}}\) |
| \(b_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(\frac {1}{36}\) |
| \(b_{4}\) | \(\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )}\) | \(\frac {29}{1440}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {12 r^{5}+132 r^{4}+549 r^{3}+1057 r^{2}+920 r +292}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right ) \left (2 r^{2}+19 r +44\right )} \] Which for the root \(r = -{\frac {1}{2}}\) becomes \[ b_{5}=-{\frac {71}{50400}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{r \left (2 r +3\right )}\) | \(-1\) |
| \(b_{2}\) | \(\frac {-2 r^{2}-r +1}{4 r^{3}+16 r^{2}+15 r}\) | \(-{\frac {1}{4}}\) |
| \(b_{3}\) | \(-\frac {4 \left (1+r \right )^{2}}{8 r^{5}+76 r^{4}+262 r^{3}+389 r^{2}+210 r}\) | \(\frac {1}{36}\) |
| \(b_{4}\) | \(\frac {4 r^{5}+32 r^{4}+85 r^{3}+73 r^{2}-16 r -32}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right )}\) | \(\frac {29}{1440}\) |
| \(b_{5}\) | \(\frac {12 r^{5}+132 r^{4}+549 r^{3}+1057 r^{2}+920 r +292}{\left (2 r^{2}+15 r +27\right ) r \left (8 r^{4}+76 r^{3}+262 r^{2}+389 r +210\right ) \left (2 r^{2}+19 r +44\right )}\) | \(-{\frac {71}{50400}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= x \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}\dots \right ) \\ &= \frac {1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\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 \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+O\left (x^{6}\right )\right ) + \frac {c_{2} \left (1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{2} y^{\prime \prime }+\left (x^{3}+x \right ) y^{\prime }+\left (-x -1\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 (x +1\right ) y}{2 x^{2}}-\frac {\left (x^{2}+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} \\ {} & {} & y^{\prime \prime }+\frac {\left (x^{2}+1\right ) y^{\prime }}{2 x}-\frac {\left (x +1\right ) y}{2 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 {x^{2}+1}{2 x}, P_{3}\left (x \right )=-\frac {x +1}{2 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}}}=-\frac {1}{2} \\ {} & \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} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }+\left (-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 =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^{2}\cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y^{\prime \prime }=\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 (-1+r \right ) x^{r}+\left (a_{1} \left (3+2 r \right ) r -a_{0}\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (2 k +2 r +1\right ) \left (k +r -1\right )-a_{k -1}+a_{k -2} \left (k -2+r \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 (-1+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{1, -\frac {1}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (3+2 r \right ) r -a_{0}=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=\frac {a_{0}}{r \left (3+2 r \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 (k +r -1\right ) a_{k}+a_{k -2} k +a_{k -2} r -2 a_{k -2}-a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & 2 \left (k +\frac {5}{2}+r \right ) \left (k +1+r \right ) a_{k +2}+a_{k} \left (k +2\right )+r a_{k}-2 a_{k}-a_{k +1}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {k a_{k}+r a_{k}-a_{k +1}}{\left (2 k +5+2 r \right ) \left (k +1+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1 \\ {} & {} & a_{k +2}=-\frac {k a_{k}+a_{k}-a_{k +1}}{\left (2 k +7\right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +1}, a_{k +2}=-\frac {k a_{k}+a_{k}-a_{k +1}}{\left (2 k +7\right ) \left (k +2\right )}, a_{1}=\frac {a_{0}}{5}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {1}{2} \\ {} & {} & a_{k +2}=-\frac {k a_{k}-\frac {1}{2} a_{k}-a_{k +1}}{\left (2 k +4\right ) \left (k +\frac {1}{2}\right )} \\ \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 {k a_{k}-\frac {1}{2} a_{k}-a_{k +1}}{\left (2 k +4\right ) \left (k +\frac {1}{2}\right )}, a_{1}=-a_{0}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +1}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k -\frac {1}{2}}\right ), a_{k +2}=-\frac {k a_{k}+a_{k}-a_{k +1}}{\left (2 k +7\right ) \left (k +2\right )}, a_{1}=\frac {a_{0}}{5}, b_{k +2}=-\frac {k b_{k}-\frac {1}{2} b_{k}-b_{k +1}}{\left (2 k +4\right ) \left (k +\frac {1}{2}\right )}, b_{1}=-b_{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 -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunB ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+x*(1+x^2)*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1-x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {29}{1440} x^{4}-\frac {71}{50400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x -\frac {2}{35} x^{2}-\frac {16}{945} x^{3}+\frac {73}{20790} x^{4}+\frac {1481}{1351350} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 84
AsymptoticDSolveValue[2*x^2*y''[x]+x*(1+x^2)*y'[x]-(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {1481 x^5}{1351350}+\frac {73 x^4}{20790}-\frac {16 x^3}{945}-\frac {2 x^2}{35}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {71 x^5}{50400}+\frac {29 x^4}{1440}+\frac {x^3}{36}-\frac {x^2}{4}-x+1\right )}{\sqrt {x}} \]