Internal problem ID [1354]
Internal file name [OUTPUT/1355_Sunday_June_05_2022_02_12_36_AM_42745091/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: 2.
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, _exact, _linear, _homogeneous]]
\[ \boxed {x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (7 x^{2}+6 x +3\right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y=0} \] 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. \[ \left (2 x^{4}+x^{3}+x^{2}\right ) y^{\prime \prime }+\left (7 x^{3}+6 x^{2}+3 x \right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y = 0 \] 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 {7 x^{2}+6 x +3}{x \left (2 x^{2}+x +1\right )}\\ q(x) &= -\frac {3 x^{2}-6 x -1}{x^{2} \left (2 x^{2}+x +1\right )}\\ \end {align*}
| \(p(x)=\frac {7 x^{2}+6 x +3}{x \left (2 x^{2}+x +1\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = -\frac {1}{4}+\frac {i \sqrt {7}}{4}\) | \(\text {``regular''}\) |
| \(x = -\frac {i \sqrt {7}}{4}-\frac {1}{4}\) | \(\text {``regular''}\) |
| \(q(x)=-\frac {3 x^{2}-6 x -1}{x^{2} \left (2 x^{2}+x +1\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = -\frac {1}{4}+\frac {i \sqrt {7}}{4}\) | \(\text {``regular''}\) |
| \(x = -\frac {i \sqrt {7}}{4}-\frac {1}{4}\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \(\left [0, -\frac {1}{4}+\frac {i \sqrt {7}}{4}, -\frac {i \sqrt {7}}{4}-\frac {1}{4}, \infty \right ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (7 x^{3}+6 x^{2}+3 x \right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y = 0 \] 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} x^{2} \left (2 x^{2}+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 )+\left (7 x^{3}+6 x^{2}+3 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (-3 x^{2}+6 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 +2} 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 ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}7 x^{n +r +2} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 x^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r +2} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n}\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 }}2 x^{n +r +2} a_{n} \left (n +r \right ) \left (n +r -1\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}2 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right ) x^{n +r} \\ \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 }}7 x^{n +r +2} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}7 a_{n -2} \left (n +r -2\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} \left (n +r -1\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r +2} a_{n}\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-3 a_{n -2} x^{n +r}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}6 x^{1+n +r} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} x^{n +r} \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =2}{\sum }}2 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right ) x^{n +r}\right )+\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 }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}7 a_{n -2} \left (n +r -2\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} \left (n +r -1\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 x^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-3 a_{n -2} x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}6 a_{n -1} x^{n +r}\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 \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+3 x^{n +r} a_{n} \left (n +r \right )+a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )+3 x^{r} a_{0} r +a_{0} x^{r} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )+3 x^{r} r +x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (r +1\right )^{2} x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ \left (r +1\right )^{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= -1\\ r_2 &= -1 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (r +1\right )^{2} x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([-1, -1]\).
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, \(r = -1\), Eqs (1A,1B) become \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n -1}\\ 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 -1}\right ) \end {align*}
We start by finding the first solution \(y_{1}\left (x \right )\). Eq (2B) derived above is now used to find all \(a_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(a_{0}\) is arbitrary and taken as \(a_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ a_{1} = \frac {-r -3}{2+r} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} 2 a_{n -2} \left (n +r -2\right ) \left (n -3+r \right )+a_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+a_{n} \left (n +r \right ) \left (n +r -1\right )+7 a_{n -2} \left (n +r -2\right )+6 a_{n -1} \left (n +r -1\right )+3 a_{n} \left (n +r \right )-3 a_{n -2}+6 a_{n -1}+a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {2 n a_{n -2}+n a_{n -1}+2 r a_{n -2}+r a_{n -1}-5 a_{n -2}+2 a_{n -1}}{1+n +r}\tag {4} \] Which for the root \(r = -1\) becomes \[ a_{n} = \frac {\left (-2 a_{n -2}-a_{n -1}\right ) n +7 a_{n -2}-a_{n -1}}{n}\tag {5} \] 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.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )} \] Which for the root \(r = -1\) becomes \[ a_{2}={\frac {9}{2}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )} \] Which for the root \(r = -1\) becomes \[ a_{3}=-{\frac {20}{3}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
| \(a_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = -1\) becomes \[ a_{4}={\frac {173}{24}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
| \(a_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) |
| \(a_{4}\) | \(\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {173}{24}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {-5 r^{5}-37 r^{4}+72 r^{3}+941 r^{2}+1534 r +139}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )} \] Which for the root \(r = -1\) becomes \[ a_{5}=-{\frac {93}{20}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
| \(a_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) |
| \(a_{4}\) | \(\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {173}{24}\) |
| \(a_{5}\) | \(\frac {-5 r^{5}-37 r^{4}+72 r^{3}+941 r^{2}+1534 r +139}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {93}{20}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {7 r^{6}+154 r^{5}+1085 r^{4}+2409 r^{3}-2291 r^{2}-12771 r -9428}{\left (7+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )} \] Which for the root \(r = -1\) becomes \[ a_{6}=-{\frac {419}{720}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
| \(a_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) |
| \(a_{4}\) | \(\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {173}{24}\) |
| \(a_{5}\) | \(\frac {-5 r^{5}-37 r^{4}+72 r^{3}+941 r^{2}+1534 r +139}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {93}{20}}\) |
| \(a_{6}\) | \(\frac {7 r^{6}+154 r^{5}+1085 r^{4}+2409 r^{3}-2291 r^{2}-12771 r -9428}{\left (7+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {419}{720}}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {3 r^{7}-28 r^{6}-1449 r^{5}-13381 r^{4}-48637 r^{3}-61453 r^{2}+24528 r +76095}{\left (8+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )} \] Which for the root \(r = -1\) becomes \[ a_{7}={\frac {6697}{1260}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) |
| \(a_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) |
| \(a_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) |
| \(a_{4}\) | \(\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {173}{24}\) |
| \(a_{5}\) | \(\frac {-5 r^{5}-37 r^{4}+72 r^{3}+941 r^{2}+1534 r +139}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {93}{20}}\) |
| \(a_{6}\) | \(\frac {7 r^{6}+154 r^{5}+1085 r^{4}+2409 r^{3}-2291 r^{2}-12771 r -9428}{\left (7+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {419}{720}}\) |
| \(a_{7}\) | \(\frac {3 r^{7}-28 r^{6}-1449 r^{5}-13381 r^{4}-48637 r^{3}-61453 r^{2}+24528 r +76095}{\left (8+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(\frac {6697}{1260}\) |
Using the above table, then the first solution \(y_{1}\left (x \right )\) is \begin{align*} y_{1}\left (x \right )&= \frac {1}{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}+a_{7} x^{7}+a_{8} x^{8}\dots \right ) \\ &= \frac {1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )}{x} \\ \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 \(r = -1\). The above table for \(a_{n ,r}\) is used for this purpose. Computing the derivatives gives the following table
| \(n\) | \(b_{n ,r}\) | \(a_{n}\) | \(b_{n ,r} = \frac {d}{d r}a_{n ,r}\) | \(b_{n}\left (r =-1\right )\) |
| \(b_{0}\) | \(1\) | \(1\) | N/A since \(b_{n}\) starts from 1 | N/A |
| \(b_{1}\) | \(\frac {-r -3}{2+r}\) | \(-2\) | \(\frac {1}{\left (2+r \right )^{2}}\) | \(1\) |
| \(b_{2}\) | \(\frac {-r^{2}+4 r +14}{\left (2+r \right ) \left (r +3\right )}\) | \(\frac {9}{2}\) | \(\frac {-9 r^{2}-40 r -46}{\left (2+r \right )^{2} \left (r +3\right )^{2}}\) | \(-{\frac {15}{4}}\) |
| \(b_{3}\) | \(\frac {3 r^{3}+14 r^{2}-10 r -61}{\left (4+r \right ) \left (2+r \right ) \left (r +3\right )}\) | \(-{\frac {20}{3}}\) | \(\frac {13 r^{4}+176 r^{3}+853 r^{2}+1770 r +1346}{\left (2+r \right )^{2} \left (r +3\right )^{2} \left (4+r \right )^{2}}\) | \(\frac {133}{18}\) |
| \(b_{4}\) | \(\frac {-r^{4}-29 r^{3}-134 r^{2}-81 r +198}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right )}\) | \(\frac {173}{24}\) | \(\frac {15 r^{6}+126 r^{5}-402 r^{4}-7936 r^{3}-33641 r^{2}-60276 r -40212}{\left (2+r \right )^{2} \left (r +3\right )^{2} \left (4+r \right )^{2} \left (5+r \right )^{2}}\) | \(-{\frac {3077}{288}}\) |
| \(b_{5}\) | \(\frac {-5 r^{5}-37 r^{4}+72 r^{3}+941 r^{2}+1534 r +139}{\left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {93}{20}}\) | \(\frac {-63 r^{8}-1694 r^{7}-18698 r^{6}-107576 r^{5}-330714 r^{4}-442884 r^{3}+183569 r^{2}+1193800 r +959364}{\left (2+r \right )^{2} \left (r +3\right )^{2} \left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2}}\) | \(\frac {4217}{400}\) |
| \(b_{6}\) | \(\frac {7 r^{6}+154 r^{5}+1085 r^{4}+2409 r^{3}-2291 r^{2}-12771 r -9428}{\left (7+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right )}\) | \(-{\frac {419}{720}}\) | \(\frac {35 r^{10}+1960 r^{9}+43873 r^{8}+534810 r^{7}+3984324 r^{6}+19020134 r^{5}+58697106 r^{4}+114204974 r^{3}+130307976 r^{2}+73147744 r +11322144}{\left (7+r \right )^{2} \left (2+r \right )^{2} \left (r +3\right )^{2} \left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2}}\) | \(-{\frac {70949}{14400}}\) |
| \(b_{7}\) | \(\frac {3 r^{7}-28 r^{6}-1449 r^{5}-13381 r^{4}-48637 r^{3}-61453 r^{2}+24528 r +76095}{\left (8+r \right ) \left (2+r \right ) \left (r +3\right ) \left (4+r \right ) \left (5+r \right ) \left (6+r \right ) \left (7+r \right )}\) | \(\frac {6697}{1260}\) | \(\frac {133 r^{12}+5964 r^{11}+112775 r^{10}+1126906 r^{9}+5604900 r^{8}+544946 r^{7}-182629895 r^{6}-1287237756 r^{5}-4807375225 r^{4}-11024611260 r^{3}-15543973032 r^{2}-12391573320 r -4281675120}{\left (8+r \right )^{2} \left (2+r \right )^{2} \left (r +3\right )^{2} \left (4+r \right )^{2} \left (5+r \right )^{2} \left (6+r \right )^{2} \left (7+r \right )^{2}}\) | \(-{\frac {125221}{29400}}\) |
The above table gives all values of \(b_{n}\) needed. Hence the second solution is \begin{align*} y_{2}\left (x \right )&=y_{1}\left (x \right ) \ln \left (x \right )+b_{0}+b_{1} x +b_{2} x^{2}+b_{3} x^{3}+b_{4} x^{4}+b_{5} x^{5}+b_{6} x^{6}+b_{7} x^{7}+b_{8} x^{8}\dots \\ &= \frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{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 ) \\ &= \frac {c_{1} \left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right )}{x} + c_{2} \left (\frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= \frac {c_{1} \left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} \left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} \left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (7 x^{3}+6 x^{2}+3 x \right ) y^{\prime }+\left (-3 x^{2}+6 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 (3 x^{2}-6 x -1\right ) y}{x^{2} \left (2 x^{2}+x +1\right )}-\frac {\left (7 x^{2}+6 x +3\right ) y^{\prime }}{x \left (2 x^{2}+x +1\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 (7 x^{2}+6 x +3\right ) y^{\prime }}{x \left (2 x^{2}+x +1\right )}-\frac {\left (3 x^{2}-6 x -1\right ) y}{x^{2} \left (2 x^{2}+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 {7 x^{2}+6 x +3}{x \left (2 x^{2}+x +1\right )}, P_{3}\left (x \right )=-\frac {3 x^{2}-6 x -1}{x^{2} \left (2 x^{2}+x +1\right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=3 \\ {} & \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}}}=1 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (7 x^{2}+6 x +3\right ) y^{\prime }+\left (-3 x^{2}+6 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..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..4 \\ {} & {} & x^{m}\cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (1+r \right )^{2} x^{r}+\left (a_{1} \left (2+r \right )^{2}+a_{0} \left (3+r \right ) \left (2+r \right )\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k +r +1\right )^{2}+a_{k -1} \left (k +r +2\right ) \left (k +r +1\right )+a_{k -2} \left (k +r +1\right ) \left (2 k -5+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+r \right )^{2}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r =-1 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (2+r \right )^{2}+a_{0} \left (3+r \right ) \left (2+r \right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {\left (3+r \right ) a_{0}}{2+r} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (\left (a_{k}+2 a_{k -2}+a_{k -1}\right ) k +\left (a_{k}+2 a_{k -2}+a_{k -1}\right ) r +a_{k}-5 a_{k -2}+2 a_{k -1}\right ) \left (k +r +1\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (a_{k +2}+2 a_{k}+a_{k +1}\right ) \left (k +2\right )+\left (a_{k +2}+2 a_{k}+a_{k +1}\right ) r +a_{k +2}-5 a_{k}+2 a_{k +1}\right ) \left (k +r +3\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {2 k a_{k}+k a_{k +1}+2 r a_{k}+r a_{k +1}-a_{k}+4 a_{k +1}}{k +r +3} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-1 \\ {} & {} & a_{k +2}=-\frac {2 k a_{k}+k a_{k +1}-3 a_{k}+3 a_{k +1}}{k +2} \\ \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 {2 k a_{k}+k a_{k +1}-3 a_{k}+3 a_{k +1}}{k +2}, a_{1}=-2 a_{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)] One independent solution has integrals. Trying a hypergeometric solution free of integrals... -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius No hypergeometric solution was found. <- linear_1 successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 81
Order:=8; dsolve(x^2*(1+x+2*x^2)*diff(y(x),x$2)+x*(3+6*x+7*x^2)*diff(y(x),x)+(1+6*x-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {9}{2} x^{2}-\frac {20}{3} x^{3}+\frac {173}{24} x^{4}-\frac {93}{20} x^{5}-\frac {419}{720} x^{6}+\frac {6697}{1260} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {15}{4} x^{2}+\frac {133}{18} x^{3}-\frac {3077}{288} x^{4}+\frac {4217}{400} x^{5}-\frac {70949}{14400} x^{6}-\frac {125221}{29400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 162
AsymptoticDSolveValue[x^2*(1+x+2*x^2)*y''[x]+x*(3+6*x+7*x^2)*y'[x]+(1+6*x-3*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right )}{x}+c_2 \left (\frac {-\frac {125221 x^7}{29400}-\frac {70949 x^6}{14400}+\frac {4217 x^5}{400}-\frac {3077 x^4}{288}+\frac {133 x^3}{18}-\frac {15 x^2}{4}+x}{x}+\frac {\left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right ) \log (x)}{x}\right ) \]