Internal problem ID [2395]
Internal file name [OUTPUT/2395_Tuesday_February_27_2024_08_36_51_AM_79605024/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: 20.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Difference not integer"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (4+x \right ) x^{2} y^{\prime \prime }+x \left (x -1\right ) y^{\prime }+y=0} \] 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. \[ \left (x^{3}+4 x^{2}\right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }+y = 0 \] 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 -1}{x \left (4+x \right )}\\ q(x) &= \frac {1}{x^{2} \left (4+x \right )}\\ \end {align*}
| \(p(x)=\frac {x -1}{x \left (4+x \right )}\) | |
| singularity | type |
| \(x = -4\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``regular''}\) |
| \(q(x)=\frac {1}{x^{2} \left (4+x \right )}\) | |
| singularity | type |
| \(x = -4\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-4, 0, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ \left (4+x \right ) x^{2} y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }+y = 0 \] 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 (4+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 (x^{2}-x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -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 }}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 )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}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^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}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} \\ \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 }}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 )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )-x^{n +r} a_{n} \left (n +r \right )+a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{r} a_{0} r \left (-1+r \right )-x^{r} a_{0} r +a_{0} x^{r} = 0 \] Or \[ \left (4 x^{r} r \left (-1+r \right )-x^{r} r +x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (4 r^{2}-5 r +1\right ) x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 4 r^{2}-5 r +1 = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 1\\ r_2 &= {\frac {1}{4}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (4 r^{2}-5 r +1\right ) x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [1, {\frac {1}{4}}\right ]\).
Since \(r_1 - r_2 = {\frac {3}{4}}\) 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}{4}} \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 \(a_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} 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} \left (n +r \right )+a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {a_{n -1} \left (n +r -1\right )}{4 n +4 r -1}\tag {4} \] Which for the root \(r = 1\) becomes \[ a_{n} = -\frac {a_{n -1} n}{4 n +3}\tag {5} \] 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.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=-\frac {r}{3+4 r} \] Which for the root \(r = 1\) becomes \[ a_{1}=-{\frac {1}{7}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{7}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {r \left (1+r \right )}{16 r^{2}+40 r +21} \] Which for the root \(r = 1\) becomes \[ a_{2}={\frac {2}{77}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{7}}\) |
| \(a_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {2}{77}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231} \] Which for the root \(r = 1\) becomes \[ a_{3}=-{\frac {2}{385}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{7}}\) |
| \(a_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {2}{77}\) |
| \(a_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {2}{385}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465} \] Which for the root \(r = 1\) becomes \[ a_{4}={\frac {8}{7315}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{7}}\) |
| \(a_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {2}{77}\) |
| \(a_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {2}{385}}\) |
| \(a_{4}\) | \(\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465}\) | \(\frac {8}{7315}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right ) \left (4+r \right )}{1024 r^{5}+14080 r^{4}+72320 r^{3}+170720 r^{2}+180756 r +65835} \] Which for the root \(r = 1\) becomes \[ a_{5}=-{\frac {8}{33649}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{7}}\) |
| \(a_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {2}{77}\) |
| \(a_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {2}{385}}\) |
| \(a_{4}\) | \(\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465}\) | \(\frac {8}{7315}\) |
| \(a_{5}\) | \(-\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right ) \left (4+r \right )}{1024 r^{5}+14080 r^{4}+72320 r^{3}+170720 r^{2}+180756 r +65835}\) | \(-{\frac {8}{33649}}\) |
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}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+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 \(b_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} b_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+4 b_{n} \left (n +r \right ) \left (n +r -1\right )+b_{n -1} \left (n +r -1\right )-b_{n} \left (n +r \right )+b_{n} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {b_{n -1} \left (n +r -1\right )}{4 n +4 r -1}\tag {4} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{n} = -\frac {b_{n -1} \left (4 n -3\right )}{16 n}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = {\frac {1}{4}}\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ b_{1}=-\frac {r}{3+4 r} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{1}=-{\frac {1}{16}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{16}}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {r \left (1+r \right )}{16 r^{2}+40 r +21} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{2}={\frac {5}{512}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{16}}\) |
| \(b_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {5}{512}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{3}=-{\frac {15}{8192}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{16}}\) |
| \(b_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {5}{512}\) |
| \(b_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {15}{8192}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{4}={\frac {195}{524288}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{16}}\) |
| \(b_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {5}{512}\) |
| \(b_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {15}{8192}}\) |
| \(b_{4}\) | \(\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465}\) | \(\frac {195}{524288}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right ) \left (4+r \right )}{1024 r^{5}+14080 r^{4}+72320 r^{3}+170720 r^{2}+180756 r +65835} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ b_{5}=-{\frac {663}{8388608}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {r}{3+4 r}\) | \(-{\frac {1}{16}}\) |
| \(b_{2}\) | \(\frac {r \left (1+r \right )}{16 r^{2}+40 r +21}\) | \(\frac {5}{512}\) |
| \(b_{3}\) | \(-\frac {\left (1+r \right ) r \left (2+r \right )}{64 r^{3}+336 r^{2}+524 r +231}\) | \(-{\frac {15}{8192}}\) |
| \(b_{4}\) | \(\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}{256 r^{4}+2304 r^{3}+7136 r^{2}+8784 r +3465}\) | \(\frac {195}{524288}\) |
| \(b_{5}\) | \(-\frac {r \left (1+r \right ) \left (2+r \right ) \left (3+r \right ) \left (4+r \right )}{1024 r^{5}+14080 r^{4}+72320 r^{3}+170720 r^{2}+180756 r +65835}\) | \(-{\frac {663}{8388608}}\) |
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 ) \\ &= x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+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} x \left (1-\frac {x}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+O\left (x^{6}\right )\right ) + c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x \left (1-\frac {x}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x \left (1-\frac {x}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x \left (1-\frac {x}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (4+x \right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }+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 {y}{x^{2} \left (4+x \right )}-\frac {\left (x -1\right ) y^{\prime }}{x \left (4+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 (x -1\right ) y^{\prime }}{x \left (4+x \right )}+\frac {y}{x^{2} \left (4+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 {x -1}{x \left (4+x \right )}, P_{3}\left (x \right )=\frac {1}{x^{2} \left (4+x \right )}\right ] \\ {} & \circ & \left (4+x \right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-4 \\ {} & {} & \left (\left (4+x \right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-4}}}=\frac {5}{4} \\ {} & \circ & \left (4+x \right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-4 \\ {} & {} & \left (\left (4+x \right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-4}}}=0 \\ {} & \circ & x =-4\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}=-4 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (4+x \right ) y^{\prime \prime }+x \left (x -1\right ) y^{\prime }+y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -4\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (u^{3}-8 u^{2}+16 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (u^{2}-9 u +20\right ) \left (\frac {d}{d u}y \left (u \right )\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 \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+4 r \right ) u^{-1+r}+\left (4 a_{1} \left (1+r \right ) \left (5+4 r \right )-a_{0} \left (8 r^{2}+r -1\right )\right ) u^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (4 a_{k +1} \left (k +1+r \right ) \left (4 k +5+4 r \right )-a_{k} \left (8 k^{2}+16 k r +8 r^{2}+k +r -1\right )+a_{k -1} \left (k +r -1\right )^{2}\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+4 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, -\frac {1}{4}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & 4 a_{1} \left (1+r \right ) \left (5+4 r \right )-a_{0} \left (8 r^{2}+r -1\right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k -1} \left (k +r -1\right )^{2}+16 \left (k +1+r \right ) \left (k +\frac {5}{4}+r \right ) a_{k +1}-a_{k} \left (8 k^{2}+16 k r +8 r^{2}+k +r -1\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k} \left (k +r \right )^{2}+16 \left (k +2+r \right ) \left (k +\frac {9}{4}+r \right ) a_{k +2}-a_{k +1} \left (8 \left (k +1\right )^{2}+16 \left (k +1\right ) r +8 r^{2}+k +r \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}+2 k r a_{k}-16 k r a_{k +1}+r^{2} a_{k}-8 r^{2} a_{k +1}-17 k a_{k +1}-17 r a_{k +1}-8 a_{k +1}}{4 \left (k +2+r \right ) \left (4 k +9+4 r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-17 k a_{k +1}-8 a_{k +1}}{4 \left (k +2\right ) \left (4 k +9\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 {k^{2} a_{k}-8 k^{2} a_{k +1}-17 k a_{k +1}-8 a_{k +1}}{4 \left (k +2\right ) \left (4 k +9\right )}, 20 a_{1}+a_{0}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =4+x \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (4+x \right )^{k}, a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-17 k a_{k +1}-8 a_{k +1}}{4 \left (k +2\right ) \left (4 k +9\right )}, 20 a_{1}+a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {1}{4} \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-\frac {1}{2} k a_{k}-13 k a_{k +1}+\frac {1}{16} a_{k}-\frac {17}{4} a_{k +1}}{4 \left (k +\frac {7}{4}\right ) \left (4 k +8\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {1}{4} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {1}{4}}, a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-\frac {1}{2} k a_{k}-13 k a_{k +1}+\frac {1}{16} a_{k}-\frac {17}{4} a_{k +1}}{4 \left (k +\frac {7}{4}\right ) \left (4 k +8\right )}, 12 a_{1}+\frac {3 a_{0}}{4}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =4+x \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (4+x \right )^{k -\frac {1}{4}}, a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-\frac {1}{2} k a_{k}-13 k a_{k +1}+\frac {1}{16} a_{k}-\frac {17}{4} a_{k +1}}{4 \left (k +\frac {7}{4}\right ) \left (4 k +8\right )}, 12 a_{1}+\frac {3 a_{0}}{4}=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (4+x \right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (4+x \right )^{k -\frac {1}{4}}\right ), a_{k +2}=-\frac {k^{2} a_{k}-8 k^{2} a_{k +1}-17 k a_{k +1}-8 a_{k +1}}{4 \left (k +2\right ) \left (4 k +9\right )}, 20 a_{1}+a_{0}=0, b_{k +2}=-\frac {k^{2} b_{k}-8 k^{2} b_{k +1}-\frac {1}{2} k b_{k}-13 k b_{k +1}+\frac {1}{16} b_{k}-\frac {17}{4} b_{k +1}}{4 \left (k +\frac {7}{4}\right ) \left (4 k +8\right )}, 12 b_{1}+\frac {3 b_{0}}{4}=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 <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form for at least one hypergeometric solution is achieved - returning with no uncomputed integrals <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
Order:=6; dsolve((4+x)*x^2*diff(y(x),x$2)+x*(x-1)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{16} x +\frac {5}{512} x^{2}-\frac {15}{8192} x^{3}+\frac {195}{524288} x^{4}-\frac {663}{8388608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {1}{7} x +\frac {2}{77} x^{2}-\frac {2}{385} x^{3}+\frac {8}{7315} x^{4}-\frac {8}{33649} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 86
AsymptoticDSolveValue[(4+x)*x^2*y''[x]+x*(x-1)*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (-\frac {8 x^5}{33649}+\frac {8 x^4}{7315}-\frac {2 x^3}{385}+\frac {2 x^2}{77}-\frac {x}{7}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {663 x^5}{8388608}+\frac {195 x^4}{524288}-\frac {15 x^3}{8192}+\frac {5 x^2}{512}-\frac {x}{16}+1\right ) \]