Internal problem ID [6925]
Internal file name [OUTPUT/6168_Tuesday_August_09_2022_05_24_01_AM_88069339/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 9.
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]]
\[ \boxed {2 x \left (x +3\right ) y^{\prime \prime }-3 \left (1+x \right ) y^{\prime }+2 y=0} \] 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. \[ \left (2 x^{2}+6 x \right ) y^{\prime \prime }+\left (-3 x -3\right ) y^{\prime }+2 y = 0 \] 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 {3 \left (1+x \right )}{2 x \left (x +3\right )}\\ q(x) &= \frac {1}{x \left (x +3\right )}\\ \end {align*}
| \(p(x)=-\frac {3 \left (1+x \right )}{2 x \left (x +3\right )}\) | |
| singularity | type |
| \(x = -3\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``regular''}\) |
| \(q(x)=\frac {1}{x \left (x +3\right )}\) | |
| singularity | type |
| \(x = -3\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-3, 0, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 x \left (x +3\right ) y^{\prime \prime }+\left (-3 x -3\right ) y^{\prime }+2 y = 0 \] 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 (x +3\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 (-3 x -3\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 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 }}2 x^{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 -1} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-3 a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} x^{n +r -1} \\ \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 =1}{\sum }}2 a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-3 a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 6 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )-3 \left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ 6 x^{-1+r} a_{0} r \left (-1+r \right )-3 r a_{0} x^{-1+r} = 0 \] Or \[ \left (6 x^{-1+r} r \left (-1+r \right )-3 r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (6 r^{2}-9 r \right ) x^{-1+r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 6 r^{2}-9 r = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {3}{2}}\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (6 r^{2}-9 r \right ) x^{-1+r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {3}{2}}, 0\right ]\).
Since \(r_1 - r_2 = {\frac {3}{2}}\) is not an integer, then we can construct two linearly independent solutions \begin {align*} y_{1}\left (x \right ) &= x^{r_{1}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right )\\ y_{2}\left (x \right ) &= x^{r_{2}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n}\right ) \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +\frac {3}{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 \(a_{0} = 1\). 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 )+6 a_{n} \left (n +r \right ) \left (n +r -1\right )-3 a_{n -1} \left (n +r -1\right )-3 a_{n} \left (n +r \right )+2 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {\left (n +r -3\right ) a_{n -1}}{3 \left (n +r \right )}\tag {4} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{n} = \frac {a_{n -1} \left (-2 n +3\right )}{6 n +9}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {3}{2}}\) 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 \[ a_{1}=\frac {2-r}{3+3 r} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{1}={\frac {1}{15}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{2}=-{\frac {1}{315}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{3}={\frac {1}{2835}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
| \(a_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(\frac {1}{2835}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{4}=-{\frac {1}{18711}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
| \(a_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(\frac {1}{2835}\) |
| \(a_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(-{\frac {1}{18711}}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{5}={\frac {1}{104247}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
| \(a_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(\frac {1}{2835}\) |
| \(a_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(-{\frac {1}{18711}}\) |
| \(a_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(\frac {1}{104247}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{6}=-{\frac {1}{521235}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
| \(a_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(\frac {1}{2835}\) |
| \(a_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(-{\frac {1}{18711}}\) |
| \(a_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(\frac {1}{104247}\) |
| \(a_{6}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )}\) | \(-{\frac {1}{521235}}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=-\frac {r \left (r^{2}-3 r +2\right )}{2187 \left (6+r \right ) \left (5+r \right ) \left (7+r \right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{7}={\frac {1}{2416635}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {1}{15}\) |
| \(a_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(-{\frac {1}{315}}\) |
| \(a_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(\frac {1}{2835}\) |
| \(a_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(-{\frac {1}{18711}}\) |
| \(a_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(\frac {1}{104247}\) |
| \(a_{6}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )}\) | \(-{\frac {1}{521235}}\) |
| \(a_{7}\) | \(-\frac {r \left (r^{2}-3 r +2\right )}{2187 \left (6+r \right ) \left (5+r \right ) \left (7+r \right )}\) | \(\frac {1}{2416635}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{\frac {3}{2}} \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 ) \\ &= x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+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 \(b_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} 2 b_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+6 b_{n} \left (n +r \right ) \left (n +r -1\right )-3 b_{n -1} \left (n +r -1\right )-3 \left (n +r \right ) b_{n}+2 b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {\left (n +r -3\right ) b_{n -1}}{3 \left (n +r \right )}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = -\frac {\left (n -3\right ) b_{n -1}}{3 n}\tag {5} \] 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.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ b_{1}=\frac {2-r}{3+3 r} \] Which for the root \(r = 0\) becomes \[ b_{1}={\frac {2}{3}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {1}{9}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{3}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
| \(b_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{4}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
| \(b_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{5}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
| \(b_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(0\) |
| \(b_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(0\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{6}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
| \(b_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(0\) |
| \(b_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(0\) |
| \(b_{6}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )}\) | \(0\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=-\frac {r \left (r^{2}-3 r +2\right )}{2187 \left (6+r \right ) \left (5+r \right ) \left (7+r \right )} \] Which for the root \(r = 0\) becomes \[ b_{7}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {2-r}{3+3 r}\) | \(\frac {2}{3}\) |
| \(b_{2}\) | \(\frac {r^{2}-3 r +2}{9 \left (1+r \right ) \left (2+r \right )}\) | \(\frac {1}{9}\) |
| \(b_{3}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{27 \left (1+r \right ) \left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{81 \left (3+r \right ) \left (2+r \right ) \left (4+r \right )}\) | \(0\) |
| \(b_{5}\) | \(-\frac {\left (r^{2}-3 r +2\right ) r}{243 \left (4+r \right ) \left (3+r \right ) \left (5+r \right )}\) | \(0\) |
| \(b_{6}\) | \(\frac {\left (r^{2}-3 r +2\right ) r}{729 \left (5+r \right ) \left (4+r \right ) \left (6+r \right )}\) | \(0\) |
| \(b_{7}\) | \(-\frac {r \left (r^{2}-3 r +2\right )}{2187 \left (6+r \right ) \left (5+r \right ) \left (7+r \right )}\) | \(0\) |
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+\frac {2 x}{3}+\frac {x^{2}}{9}+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} x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+O\left (x^{8}\right )\right ) + c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{9}+O\left (x^{8}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{9}+O\left (x^{8}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{9}+O\left (x^{8}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{9}+O\left (x^{8}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x \left (x +3\right ) y^{\prime \prime }+\left (-3 x -3\right ) y^{\prime }+2 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 {3 \left (1+x \right ) y^{\prime }}{2 x \left (x +3\right )}-\frac {y}{x \left (x +3\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 {3 \left (1+x \right ) y^{\prime }}{2 x \left (x +3\right )}+\frac {y}{x \left (x +3\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 {3 \left (1+x \right )}{2 x \left (x +3\right )}, P_{3}\left (x \right )=\frac {1}{x \left (x +3\right )}\right ] \\ {} & \circ & \left (x +3\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-3 \\ {} & {} & \left (\left (x +3\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-3}}}=-1 \\ {} & \circ & \left (x +3\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-3 \\ {} & {} & \left (\left (x +3\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-3}}}=0 \\ {} & \circ & x =-3\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}=-3 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 x \left (x +3\right ) y^{\prime \prime }+\left (-3 x -3\right ) y^{\prime }+2 y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -3\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (2 u^{2}-6 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (-3 u +6\right ) \left (\frac {d}{d u}y \left (u \right )\right )+2 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..1 \\ {} & {} & 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..2 \\ {} & {} & 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} \\ {} & {} & -6 a_{0} r \left (-2+r \right ) u^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (-6 a_{k +1} \left (k +1+r \right ) \left (k +r -1\right )+a_{k} \left (2 k +2 r -1\right ) \left (k +r -2\right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -6 r \left (-2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 2\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 2 \left (k +r -2\right ) \left (k +r -\frac {1}{2}\right ) a_{k}-6 a_{k +1} \left (k +1+r \right ) \left (k +r -1\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {\left (k +r -2\right ) \left (2 k +2 r -1\right ) a_{k}}{6 \left (k +1+r \right ) \left (k +r -1\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0\hspace {3pt}\textrm {; series terminates at}\hspace {3pt} k =2 \\ {} & {} & a_{k +1}=\frac {\left (k -2\right ) \left (2 k -1\right ) a_{k}}{6 \left (k +1\right ) \left (k -1\right )} \\ \bullet & {} & \textrm {Series not valid for}\hspace {3pt} r =0\hspace {3pt}\textrm {, division by}\hspace {3pt} 0\hspace {3pt}\textrm {in the recursion relation at}\hspace {3pt} k =1 \\ {} & {} & a_{k +1}=\frac {\left (k -2\right ) \left (2 k -1\right ) a_{k}}{6 \left (k +1\right ) \left (k -1\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +1}=\frac {k \left (2 k +3\right ) a_{k}}{6 \left (k +3\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +2}, a_{k +1}=\frac {k \left (2 k +3\right ) a_{k}}{6 \left (k +3\right ) \left (k +1\right )}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +3 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +3\right )^{k +2}, a_{k +1}=\frac {k \left (2 k +3\right ) a_{k}}{6 \left (k +3\right ) \left (k +1\right )}\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: 0.032 (sec). Leaf size: 42
Order:=8; dsolve(2*x*(x+3)*diff(y(x),x$2)-3*(x+1)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {1}{15} x -\frac {1}{315} x^{2}+\frac {1}{2835} x^{3}-\frac {1}{18711} x^{4}+\frac {1}{104247} x^{5}-\frac {1}{521235} x^{6}+\frac {1}{2416635} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2}{3} x +\frac {1}{9} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 78
AsymptoticDSolveValue[2*x*(x+3)*y''[x]-3*(x+1)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^2}{9}+\frac {2 x}{3}+1\right )+c_1 \left (\frac {x^7}{2416635}-\frac {x^6}{521235}+\frac {x^5}{104247}-\frac {x^4}{18711}+\frac {x^3}{2835}-\frac {x^2}{315}+\frac {x}{15}+1\right ) x^{3/2} \]