Internal problem ID [1369]
Internal file name [OUTPUT/1370_Sunday_June_05_2022_02_13_34_AM_11378560/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: 17.
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]]
\[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y=0} \] 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. \[ \left (2 x^{3}+4 x^{2}\right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y = 0 \] 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}{4+2 x}\\ q(x) &= -\frac {x -1}{2 x^{2} \left (2+x \right )}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-2, 0, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y = 0 \] 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} 2 x^{2} \left (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 (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right ) x^{2}+\left (1-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 }}2 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 }}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 (-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 }}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 }}x^{1+n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} \left (n +r -1\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 =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 }}4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}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 (-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 \[ x^{r} \left (2 r -1\right )^{2} = 0 \] 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 \[ x^{r} \left (2 r -1\right )^{2} = 0 \] 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} 2 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 )+a_{n -1} \left (n +r -1\right )+a_{n}-a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {\left (n +r -2\right ) a_{n -1}}{-1+2 n +2 r}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = -\frac {\left (2 n -3\right ) a_{n -1}}{4 n}\tag {5} \] 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.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=\frac {1-r}{1+2 r} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{1}={\frac {1}{4}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}=-{\frac {1}{32}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) |
| \(a_{2}\) | \(\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3}\) | \(-{\frac {1}{32}}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {-r^{3}+r}{8 r^{3}+36 r^{2}+46 r +15} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {1}{128}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) |
| \(a_{2}\) | \(\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3}\) | \(-{\frac {1}{32}}\) |
| \(a_{3}\) | \(\frac {-r^{3}+r}{8 r^{3}+36 r^{2}+46 r +15}\) | \(\frac {1}{128}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {r \left (r^{2}-1\right ) \left (2+r \right )}{\left (7+2 r \right ) \left (8 r^{3}+36 r^{2}+46 r +15\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}=-{\frac {5}{2048}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) |
| \(a_{2}\) | \(\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3}\) | \(-{\frac {1}{32}}\) |
| \(a_{3}\) | \(\frac {-r^{3}+r}{8 r^{3}+36 r^{2}+46 r +15}\) | \(\frac {1}{128}\) |
| \(a_{4}\) | \(\frac {r \left (r^{2}-1\right ) \left (2+r \right )}{\left (7+2 r \right ) \left (8 r^{3}+36 r^{2}+46 r +15\right )}\) | \(-{\frac {5}{2048}}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {-r^{5}-5 r^{4}-5 r^{3}+5 r^{2}+6 r}{32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {7}{8192}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) |
| \(a_{2}\) | \(\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3}\) | \(-{\frac {1}{32}}\) |
| \(a_{3}\) | \(\frac {-r^{3}+r}{8 r^{3}+36 r^{2}+46 r +15}\) | \(\frac {1}{128}\) |
| \(a_{4}\) | \(\frac {r \left (r^{2}-1\right ) \left (2+r \right )}{\left (7+2 r \right ) \left (8 r^{3}+36 r^{2}+46 r +15\right )}\) | \(-{\frac {5}{2048}}\) |
| \(a_{5}\) | \(\frac {-r^{5}-5 r^{4}-5 r^{3}+5 r^{2}+6 r}{32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945}\) | \(\frac {7}{8192}\) |
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 (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+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 |
| \(b_{1}\) | \(\frac {1-r}{1+2 r}\) | \(\frac {1}{4}\) | \(-\frac {3}{\left (1+2 r \right )^{2}}\) | \(-{\frac {3}{4}}\) |
| \(b_{2}\) | \(\frac {r \left (-1+r \right )}{4 r^{2}+8 r +3}\) | \(-{\frac {1}{32}}\) | \(\frac {12 r^{2}+6 r -3}{\left (4 r^{2}+8 r +3\right )^{2}}\) | \(\frac {3}{64}\) |
| \(b_{3}\) | \(\frac {-r^{3}+r}{8 r^{3}+36 r^{2}+46 r +15}\) | \(\frac {1}{128}\) | \(\frac {-36 r^{4}-108 r^{3}-81 r^{2}+15}{\left (8 r^{3}+36 r^{2}+46 r +15\right )^{2}}\) | \(-{\frac {7}{768}}\) |
| \(b_{4}\) | \(\frac {r \left (r^{2}-1\right ) \left (2+r \right )}{\left (7+2 r \right ) \left (8 r^{3}+36 r^{2}+46 r +15\right )}\) | \(-{\frac {5}{2048}}\) | \(\frac {96 r^{6}+720 r^{5}+1968 r^{4}+2340 r^{3}+966 r^{2}-210 r -210}{\left (7+2 r \right )^{2} \left (8 r^{3}+36 r^{2}+46 r +15\right )^{2}}\) | \(\frac {61}{24576}\) |
| \(b_{5}\) | \(\frac {-r^{5}-5 r^{4}-5 r^{3}+5 r^{2}+6 r}{32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945}\) | \(\frac {7}{8192}\) | \(-\frac {15 \left (16 r^{8}+224 r^{7}+1272 r^{6}+3752 r^{5}+6053 r^{4}+4984 r^{3}+1339 r^{2}-630 r -378\right )}{\left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )^{2}}\) | \(-{\frac {391}{491520}}\) |
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 (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+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 (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) + c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+O\left (x^{6}\right )\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+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 (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+O\left (x^{6}\right )\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-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 (x -1\right ) y}{2 x^{2} \left (2+x \right )}-\frac {y^{\prime }}{2 \left (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 {y^{\prime }}{2 \left (2+x \right )}-\frac {\left (x -1\right ) y}{2 x^{2} \left (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 {1}{2 \left (2+x \right )}, P_{3}\left (x \right )=-\frac {x -1}{2 x^{2} \left (2+x \right )}\right ] \\ {} & \circ & \left (2+x \right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-2 \\ {} & {} & \left (\left (2+x \right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-2}}}=\frac {1}{2} \\ {} & \circ & \left (2+x \right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-2 \\ {} & {} & \left (\left (2+x \right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-2}}}=0 \\ {} & \circ & x =-2\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}=-2 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -2\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (2 u^{3}-8 u^{2}+8 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (u^{2}-4 u +4\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (3-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 \left (-1+2 r \right ) u^{-1+r}+\left (4 a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (8 r^{2}-4 r -3\right )\right ) u^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (4 a_{k +1} \left (k +1+r \right ) \left (2 k +2 r +1\right )-a_{k} \left (8 k^{2}+16 k r +8 r^{2}-4 k -4 r -3\right )+a_{k -1} \left (2 k +2 r -1\right ) \left (k -2+r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 4 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} \\ {} & {} & 4 a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (8 r^{2}-4 r -3\right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 2 \left (-4 a_{k}+a_{k -1}+4 a_{k +1}\right ) k^{2}+\left (4 \left (-4 a_{k}+a_{k -1}+4 a_{k +1}\right ) r +4 a_{k}-5 a_{k -1}+12 a_{k +1}\right ) k +2 \left (-4 a_{k}+a_{k -1}+4 a_{k +1}\right ) r^{2}+\left (4 a_{k}-5 a_{k -1}+12 a_{k +1}\right ) r +3 a_{k}+2 a_{k -1}+4 a_{k +1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 2 \left (-4 a_{k +1}+a_{k}+4 a_{k +2}\right ) \left (k +1\right )^{2}+\left (4 \left (-4 a_{k +1}+a_{k}+4 a_{k +2}\right ) r +4 a_{k +1}-5 a_{k}+12 a_{k +2}\right ) \left (k +1\right )+2 \left (-4 a_{k +1}+a_{k}+4 a_{k +2}\right ) r^{2}+\left (4 a_{k +1}-5 a_{k}+12 a_{k +2}\right ) r +3 a_{k +1}+2 a_{k}+4 a_{k +2}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}+4 k r a_{k}-16 k r a_{k +1}+2 r^{2} a_{k}-8 r^{2} a_{k +1}-k a_{k}-12 k a_{k +1}-r a_{k}-12 r a_{k +1}-a_{k}-a_{k +1}}{4 \left (2 k^{2}+4 k r +2 r^{2}+7 k +7 r +6\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}-k a_{k}-12 k a_{k +1}-a_{k}-a_{k +1}}{4 \left (2 k^{2}+7 k +6\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 {2 k^{2} a_{k}-8 k^{2} a_{k +1}-k a_{k}-12 k a_{k +1}-a_{k}-a_{k +1}}{4 \left (2 k^{2}+7 k +6\right )}, 4 a_{1}+3 a_{0}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =2+x \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (2+x \right )^{k}, a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}-k a_{k}-12 k a_{k +1}-a_{k}-a_{k +1}}{4 \left (2 k^{2}+7 k +6\right )}, 4 a_{1}+3 a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}+k a_{k}-20 k a_{k +1}-a_{k}-9 a_{k +1}}{4 \left (2 k^{2}+9 k +10\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {1}{2}}, a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}+k a_{k}-20 k a_{k +1}-a_{k}-9 a_{k +1}}{4 \left (2 k^{2}+9 k +10\right )}, 12 a_{1}+3 a_{0}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =2+x \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (2+x \right )^{k +\frac {1}{2}}, a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{k +1}+k a_{k}-20 k a_{k +1}-a_{k}-9 a_{k +1}}{4 \left (2 k^{2}+9 k +10\right )}, 12 a_{1}+3 a_{0}=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (2+x \right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (2+x \right )^{k +\frac {1}{2}}\right ), a_{k +2}=-\frac {2 k^{2} a_{k}-8 k^{2} a_{1+k}-k a_{k}-12 k a_{1+k}-a_{k}-a_{1+k}}{4 \left (2 k^{2}+7 k +6\right )}, 4 a_{1}+3 a_{0}=0, b_{k +2}=-\frac {2 k^{2} b_{k}-8 k^{2} b_{1+k}+k b_{k}-20 k b_{1+k}-b_{k}-9 b_{1+k}}{4 \left (2 k^{2}+9 k +10\right )}, 12 b_{1}+3 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) Group is reducible, not completely reducible <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
Order:=6; dsolve(2*x^2*(2+x)*diff(y(x),x$2)+x^2*diff(y(x),x)+(1-x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x -\frac {1}{32} x^{2}+\frac {1}{128} x^{3}-\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {3}{4} x +\frac {3}{64} x^{2}-\frac {7}{768} x^{3}+\frac {61}{24576} x^{4}-\frac {391}{491520} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 134
AsymptoticDSolveValue[2*x^2*(2+x)*y''[x]+x^2*y'[x]+(1-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {391 x^5}{491520}+\frac {61 x^4}{24576}-\frac {7 x^3}{768}+\frac {3 x^2}{64}-\frac {3 x}{4}\right )+\sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right ) \log (x)\right ) \]