Internal problem ID [7002]
Internal file name [OUTPUT/6245_Thursday_August_18_2022_07_11_33_AM_88339133/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises
page 391
Problem number: 10.
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]]
\[ \boxed {2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-\left (1+x \right ) y=0} \] 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. \[ 2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+\left (-x -1\right ) y = 0 \] 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 -1}{2 x}\\ q(x) &= -\frac {1+x}{2 x}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0]\)
Irregular singular points : \([\infty ]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+\left (-x -1\right ) y = 0 \] 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 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+\left (1-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 -1} a_{n} \left (n +r \right ) \left (n +r -1\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 }}\left (n +r \right ) a_{n} x^{n +r -1}\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 \(n +r -1\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r -1}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{1+n +r} a_{n}\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-a_{n -2} x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} x^{n +r}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r -1}\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 \(n +r -1\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-a_{n -2} x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+\left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ 2 x^{-1+r} a_{0} r \left (-1+r \right )+r a_{0} x^{-1+r} = 0 \] Or \[ \left (2 x^{-1+r} r \left (-1+r \right )+r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ r \,x^{-1+r} \left (-1+2 r \right ) = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 2 r^{2}-r = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {1}{2}}\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ r \,x^{-1+r} \left (-1+2 r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {1}{2}}, 0\right ]\).
Since \(r_1 - r_2 = {\frac {1}{2}}\) 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^{n +\frac {1}{2}}\\ y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n} \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 \[ a_{1} = \frac {1}{2 r +1} \] 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 -1} \left (n +r -1\right )+a_{n} \left (n +r \right )-a_{n -2}-a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {n a_{n -1}+r a_{n -1}+a_{n -2}}{2 n^{2}+4 n r +2 r^{2}-n -r}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = \frac {2 n a_{n -1}+2 a_{n -2}+a_{n -1}}{4 n^{2}+2 n}\tag {5} \] 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.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}={\frac {9}{40}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {103}{1680}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
| \(a_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {103}{1680}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}={\frac {187}{13440}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
| \(a_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {103}{1680}\) |
| \(a_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {187}{13440}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {247}{98560}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
| \(a_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {103}{1680}\) |
| \(a_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {187}{13440}\) |
| \(a_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {247}{98560}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{6}={\frac {17861}{46126080}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
| \(a_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {103}{1680}\) |
| \(a_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {187}{13440}\) |
| \(a_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {247}{98560}\) |
| \(a_{6}\) | \(\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {17861}{46126080}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {85 r^{6}+2013 r^{5}+19036 r^{4}+91329 r^{3}+231916 r^{2}+290301 r +135135}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+27 r +91\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{7}={\frac {23767}{461260800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {1}{2 r +1}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {9}{40}\) |
| \(a_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {103}{1680}\) |
| \(a_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {187}{13440}\) |
| \(a_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {247}{98560}\) |
| \(a_{6}\) | \(\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {17861}{46126080}\) |
| \(a_{7}\) | \(\frac {85 r^{6}+2013 r^{5}+19036 r^{4}+91329 r^{3}+231916 r^{2}+290301 r +135135}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+27 r +91\right )}\) | \(\frac {23767}{461260800}\) |
Using the above table, then the 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}+a_{7} x^{7}+a_{8} x^{8}\dots \right ) \\ &= \sqrt {x}\, \left (1+\frac {x}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\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 \[ b_{1} = \frac {1}{2 r +1} \] 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 -1} \left (n +r -1\right )+\left (n +r \right ) b_{n}-b_{n -2}-b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = \frac {n b_{n -1}+r b_{n -1}+b_{n -2}}{2 n^{2}+4 n r +2 r^{2}-n -r}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = \frac {n b_{n -1}+b_{n -2}}{n \left (2 n -1\right )}\tag {5} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = 0\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90} \] Which for the root \(r = 0\) becomes \[ b_{3}={\frac {1}{6}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {1}{6}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )} \] Which for the root \(r = 0\) becomes \[ b_{4}={\frac {1}{24}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {1}{6}\) |
| \(b_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {1}{24}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )} \] Which for the root \(r = 0\) becomes \[ b_{5}={\frac {1}{120}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {1}{6}\) |
| \(b_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {1}{24}\) |
| \(b_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {1}{120}\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+11 r +15\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+19 r +45\right ) \left (2 r^{2}+23 r +66\right )} \] Which for the root \(r = 0\) becomes \[ b_{6}={\frac {1}{720}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {1}{6}\) |
| \(b_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {1}{24}\) |
| \(b_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {1}{120}\) |
| \(b_{6}\) | \(\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+11 r +15\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+19 r +45\right ) \left (2 r^{2}+23 r +66\right )}\) | \(\frac {1}{720}\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=\frac {85 r^{6}+2013 r^{5}+19036 r^{4}+91329 r^{3}+231916 r^{2}+290301 r +135135}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+27 r +91\right )} \] Which for the root \(r = 0\) becomes \[ b_{7}={\frac {1}{5040}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {1}{2 r +1}\) | \(1\) |
| \(b_{2}\) | \(\frac {3 r +3}{4 r^{3}+16 r^{2}+19 r +6}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {5 r^{2}+19 r +15}{8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90}\) | \(\frac {1}{6}\) |
| \(b_{4}\) | \(\frac {11 r^{3}+78 r^{2}+169 r +105}{\left (2 r^{2}+15 r +28\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+11 r +15\right )}\) | \(\frac {1}{24}\) |
| \(b_{5}\) | \(\frac {21 r^{4}+246 r^{3}+1014 r^{2}+1707 r +945}{\left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right )}\) | \(\frac {1}{120}\) |
| \(b_{6}\) | \(\frac {43 r^{5}+737 r^{4}+4805 r^{3}+14722 r^{2}+20787 r +10395}{\left (2 r^{2}+11 r +15\right ) \left (4 r^{3}+16 r^{2}+19 r +6\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+19 r +45\right ) \left (2 r^{2}+23 r +66\right )}\) | \(\frac {1}{720}\) |
| \(b_{7}\) | \(\frac {85 r^{6}+2013 r^{5}+19036 r^{4}+91329 r^{3}+231916 r^{2}+290301 r +135135}{\left (2 r^{2}+23 r +66\right ) \left (2 r^{2}+19 r +45\right ) \left (8 r^{5}+76 r^{4}+274 r^{3}+461 r^{2}+351 r +90\right ) \left (2 r^{2}+15 r +28\right ) \left (2 r^{2}+27 r +91\right )}\) | \(\frac {1}{5040}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\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 \\ &= 1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\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}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\right )\right ) + c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x y^{\prime \prime }+\left (1-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 (1+x \right ) y}{2 x}+\frac {\left (x -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 -1\right ) y^{\prime }}{2 x}-\frac {\left (1+x \right ) y}{2 x}=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 -1}{2 x}, P_{3}\left (x \right )=-\frac {1+x}{2 x}\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}}}=0 \\ {} & \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 y^{\prime \prime }+\left (1-x \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 =0..1 \\ {} & {} & 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 \cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & x \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} r \left (-1+2 r \right ) x^{-1+r}+\left (a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (1+r \right )\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k +1} \left (k +1+r \right ) \left (2 k +1+2 r \right )-a_{k} \left (k +1+r \right )-a_{k -1}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 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} \\ {} & {} & a_{1} \left (1+r \right ) \left (1+2 r \right )-a_{0} \left (1+r \right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k +1} \left (k +1+r \right ) \left (2 k +1+2 r \right )-a_{k} \left (k +1+r \right )-a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k +2} \left (k +2+r \right ) \left (2 k +3+2 r \right )-a_{k +1} \left (k +2+r \right )-a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {k a_{k +1}+r a_{k +1}+a_{k}+2 a_{k +1}}{\left (k +2+r \right ) \left (2 k +3+2 r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=\frac {k a_{k +1}+a_{k}+2 a_{k +1}}{\left (k +2\right ) \left (2 k +3\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +2}=\frac {k a_{k +1}+a_{k}+2 a_{k +1}}{\left (k +2\right ) \left (2 k +3\right )}, a_{1}-a_{0}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +2}=\frac {k a_{k +1}+a_{k}+\frac {5}{2} a_{k +1}}{\left (k +\frac {5}{2}\right ) \left (2 k +4\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 +1}+a_{k}+\frac {5}{2} a_{k +1}}{\left (k +\frac {5}{2}\right ) \left (2 k +4\right )}, 3 a_{1}-\frac {3 a_{0}}{2}=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {1}{2}}\right ), a_{k +2}=\frac {k a_{k +1}+a_{k}+2 a_{k +1}}{\left (k +2\right ) \left (2 k +3\right )}, a_{1}-a_{0}=0, b_{k +2}=\frac {k b_{k +1}+b_{k}+\frac {5}{2} b_{k +1}}{\left (k +\frac {5}{2}\right ) \left (2 k +4\right )}, 3 b_{1}-\frac {3 b_{0}}{2}=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 successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form is not straightforward to achieve - returning special function solution free of uncomputed integrals <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 52
Order:=8; dsolve(2*x*diff(y(x),x$2)+(1-x)*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x +\frac {9}{40} x^{2}+\frac {103}{1680} x^{3}+\frac {187}{13440} x^{4}+\frac {247}{98560} x^{5}+\frac {17861}{46126080} x^{6}+\frac {23767}{461260800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 75
AsymptoticDSolveValue[x*(x-2)^2*y''[x]-2*(x-2)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]