Internal problem ID [6920]
Internal file name [OUTPUT/6163_Tuesday_August_09_2022_05_23_49_AM_59709833/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: 4.
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
[[_Emden, _Fowler]]
\[ \boxed {4 x y^{\prime \prime }+3 y^{\prime }+3 y=0} \] 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. \[ 4 x y^{\prime \prime }+3 y^{\prime }+3 y = 0 \] 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}{4 x}\\ q(x) &= \frac {3}{4 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 \[ 4 x y^{\prime \prime }+3 y^{\prime }+3 y = 0 \] 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} 4 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+3 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+3 \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 }}4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 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 }}3 a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}3 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 =0}{\sum }}4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}3 a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+3 \left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{-1+r} a_{0} r \left (-1+r \right )+3 r a_{0} x^{-1+r} = 0 \] Or \[ \left (4 x^{-1+r} r \left (-1+r \right )+3 r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ r \,x^{-1+r} \left (-1+4 r \right ) = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 4 r^{2}-r = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {1}{4}}\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ r \,x^{-1+r} \left (-1+4 r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {1}{4}}, 0\right ]\).
Since \(r_1 - r_2 = {\frac {1}{4}}\) 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^{n +\frac {1}{4}}\\ 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} 4 a_{n} \left (n +r \right ) \left (n +r -1\right )+3 a_{n} \left (n +r \right )+3 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {3 a_{n -1}}{4 n^{2}+8 n r +4 r^{2}-n -r}\tag {4} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{n} = -\frac {3 a_{n -1}}{4 n^{2}+n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {1}{4}}\) 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 \[ a_{1}=-\frac {3}{4 r^{2}+7 r +3} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{1}=-{\frac {3}{5}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{2}={\frac {1}{10}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{3}=-{\frac {1}{130}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
| \(a_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {1}{130}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{4}={\frac {3}{8840}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
| \(a_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {1}{130}}\) |
| \(a_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{8840}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{5}=-{\frac {3}{309400}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
| \(a_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {1}{130}}\) |
| \(a_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{8840}\) |
| \(a_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {3}{309400}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{6}={\frac {3}{15470000}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
| \(a_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {1}{130}}\) |
| \(a_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{8840}\) |
| \(a_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {3}{309400}}\) |
| \(a_{6}\) | \(\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )}\) | \(\frac {3}{15470000}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=-\frac {2187}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right ) \left (4 r^{2}+55 r +189\right )} \] Which for the root \(r = {\frac {1}{4}}\) becomes \[ a_{7}=-{\frac {9}{3140410000}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-{\frac {3}{5}}\) |
| \(a_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {1}{10}\) |
| \(a_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {1}{130}}\) |
| \(a_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{8840}\) |
| \(a_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {3}{309400}}\) |
| \(a_{6}\) | \(\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )}\) | \(\frac {3}{15470000}\) |
| \(a_{7}\) | \(-\frac {2187}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right ) \left (4 r^{2}+55 r +189\right )}\) | \(-{\frac {9}{3140410000}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{\frac {1}{4}} \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 {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+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} 4 b_{n} \left (n +r \right ) \left (n +r -1\right )+3 \left (n +r \right ) b_{n}+3 b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = -\frac {3 b_{n -1}}{4 n^{2}+8 n r +4 r^{2}-n -r}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = -\frac {3 b_{n -1}}{n \left (4 n -1\right )}\tag {5} \] 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.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ b_{1}=-\frac {3}{4 r^{2}+7 r +3} \] Which for the root \(r = 0\) becomes \[ b_{1}=-1 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {3}{14}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )} \] Which for the root \(r = 0\) becomes \[ b_{3}=-{\frac {3}{154}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
| \(b_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {3}{154}}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )} \] Which for the root \(r = 0\) becomes \[ b_{4}={\frac {3}{3080}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
| \(b_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {3}{154}}\) |
| \(b_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{3080}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )} \] Which for the root \(r = 0\) becomes \[ b_{5}=-{\frac {9}{292600}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
| \(b_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {3}{154}}\) |
| \(b_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{3080}\) |
| \(b_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {9}{292600}}\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )} \] Which for the root \(r = 0\) becomes \[ b_{6}={\frac {9}{13459600}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
| \(b_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {3}{154}}\) |
| \(b_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{3080}\) |
| \(b_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {9}{292600}}\) |
| \(b_{6}\) | \(\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )}\) | \(\frac {9}{13459600}\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=-\frac {2187}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right ) \left (4 r^{2}+55 r +189\right )} \] Which for the root \(r = 0\) becomes \[ b_{7}=-{\frac {1}{94217200}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(-\frac {3}{4 r^{2}+7 r +3}\) | \(-1\) |
| \(b_{2}\) | \(\frac {9}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right )}\) | \(\frac {3}{14}\) |
| \(b_{3}\) | \(-\frac {27}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right )}\) | \(-{\frac {3}{154}}\) |
| \(b_{4}\) | \(\frac {81}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right )}\) | \(\frac {3}{3080}\) |
| \(b_{5}\) | \(-\frac {243}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right )}\) | \(-{\frac {9}{292600}}\) |
| \(b_{6}\) | \(\frac {729}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right )}\) | \(\frac {9}{13459600}\) |
| \(b_{7}\) | \(-\frac {2187}{\left (4 r^{2}+7 r +3\right ) \left (4 r^{2}+15 r +14\right ) \left (4 r^{2}+23 r +33\right ) \left (4 r^{2}+31 r +60\right ) \left (4 r^{2}+39 r +95\right ) \left (4 r^{2}+47 r +138\right ) \left (4 r^{2}+55 r +189\right )}\) | \(-{\frac {1}{94217200}}\) |
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 {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+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 {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+O\left (x^{8}\right )\right ) + c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+O\left (x^{8}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{\frac {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+O\left (x^{8}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+O\left (x^{8}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+O\left (x^{8}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x \left (\frac {d}{d x}y^{\prime }\right )+3 y^{\prime }+3 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {3 y}{4 x}-\frac {3 y^{\prime }}{4 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} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {3 y^{\prime }}{4 x}+\frac {3 y}{4 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 {3}{4 x}, P_{3}\left (x \right )=\frac {3}{4 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 {3}{4} \\ {} & \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} \\ {} & {} & 4 x \left (\frac {d}{d x}y^{\prime }\right )+3 y^{\prime }+3 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} y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\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 \left (\frac {d}{d x}y^{\prime }\right )=\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+4 r \right ) x^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +1} \left (k +1+r \right ) \left (4 k +3+4 r \right )+3 a_{k}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 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 in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 4 \left (k +1+r \right ) \left (k +\frac {3}{4}+r \right ) a_{k +1}+3 a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {3 a_{k}}{\left (k +1+r \right ) \left (4 k +3+4 r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {3 a_{k}}{\left (k +1\right ) \left (4 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 +1}=-\frac {3 a_{k}}{\left (k +1\right ) \left (4 k +3\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{4} \\ {} & {} & a_{k +1}=-\frac {3 a_{k}}{\left (k +\frac {5}{4}\right ) \left (4 k +4\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{4} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{4}}, a_{k +1}=-\frac {3 a_{k}}{\left (k +\frac {5}{4}\right ) \left (4 k +4\right )}\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}{4}}\right ), a_{k +1}=-\frac {3 a_{k}}{\left (k +1\right ) \left (4 k +3\right )}, b_{k +1}=-\frac {3 b_{k}}{\left (k +\frac {5}{4}\right ) \left (4 k +4\right )}\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)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 52
Order:=8; dsolve(4*x*diff(y(x),x$2)+3*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {3}{5} x +\frac {1}{10} x^{2}-\frac {1}{130} x^{3}+\frac {3}{8840} x^{4}-\frac {3}{309400} x^{5}+\frac {3}{15470000} x^{6}-\frac {9}{3140410000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3}{14} x^{2}-\frac {3}{154} x^{3}+\frac {3}{3080} x^{4}-\frac {9}{292600} x^{5}+\frac {9}{13459600} x^{6}-\frac {1}{94217200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 111
AsymptoticDSolveValue[4*x*y''[x]+3*y'[x]+3*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {9 x^7}{3140410000}+\frac {3 x^6}{15470000}-\frac {3 x^5}{309400}+\frac {3 x^4}{8840}-\frac {x^3}{130}+\frac {x^2}{10}-\frac {3 x}{5}+1\right )+c_2 \left (-\frac {x^7}{94217200}+\frac {9 x^6}{13459600}-\frac {9 x^5}{292600}+\frac {3 x^4}{3080}-\frac {3 x^3}{154}+\frac {3 x^2}{14}-x+1\right ) \]