Internal problem ID [6927]
Internal file name [OUTPUT/6170_Friday_August_12_2022_11_04_09_PM_9550685/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: 11.
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], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {x \left (4-x \right ) y^{\prime \prime }+\left (2-x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ \left (-x^{2}+4 x \right ) y^{\prime \prime }+\left (2-x \right ) y^{\prime }+4 y = 0 \] 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 -2}{x \left (-4+x \right )}\\ q(x) &= -\frac {4}{x \left (-4+x \right )}\\ \end {align*}
| \(p(x)=\frac {x -2}{x \left (-4+x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 4\) | \(\text {``regular''}\) |
| \(q(x)=-\frac {4}{x \left (-4+x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 4\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0, 4, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ -y^{\prime \prime } x \left (-4+x \right )+\left (2-x \right ) y^{\prime }+4 y = 0 \] 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} -\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right ) x \left (-4+x \right )+\left (2-x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{n +r -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 }}2 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 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 ) \left (n +r -1\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right ) \\ \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 }}4 a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}4 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} \moverset {\infty }{\munderset {n =1}{\sum }}\left (-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 }}4 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 }}2 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}4 a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+2 \left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{-1+r} a_{0} r \left (-1+r \right )+2 r a_{0} x^{-1+r} = 0 \] Or \[ \left (4 x^{-1+r} r \left (-1+r \right )+2 r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (4 r^{2}-2 r \right ) x^{-1+r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 4 r^{2}-2 r = 0 \] 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 \[ \left (4 r^{2}-2 r \right ) x^{-1+r} = 0 \] 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 \(a_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} -a_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+4 a_{n} \left (n +r \right ) \left (n +r -1\right )-a_{n -1} \left (n +r -1\right )+2 a_{n} \left (n +r \right )+4 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {a_{n -1} \left (n^{2}+2 n r +r^{2}-2 n -2 r -3\right )}{4 n^{2}+8 n r +4 r^{2}-2 n -2 r}\tag {4} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{n} = \frac {a_{n -1} \left (4 n^{2}-4 n -15\right )}{16 n^{2}+8 n}\tag {5} \] 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.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=\frac {r^{2}-4}{4 r^{2}+6 r +2} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{1}=-{\frac {5}{8}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{2}={\frac {7}{128}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{3}={\frac {3}{1024}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
| \(a_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(\frac {3}{1024}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{4}={\frac {11}{32768}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
| \(a_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(\frac {3}{1024}\) |
| \(a_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(\frac {11}{32768}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{5}={\frac {13}{262144}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
| \(a_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(\frac {3}{1024}\) |
| \(a_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(\frac {11}{32768}\) |
| \(a_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(\frac {13}{262144}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{6}={\frac {35}{4194304}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
| \(a_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(\frac {3}{1024}\) |
| \(a_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(\frac {11}{32768}\) |
| \(a_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(\frac {13}{262144}\) |
| \(a_{6}\) | \(\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(\frac {35}{4194304}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {\left (r^{2}+12 r +32\right ) \left (r^{2}-4\right ) r \left (-1+r \right ) \left (r +3\right )}{128 \left (2 r +13\right ) \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )} \] Which for the root \(r = {\frac {1}{2}}\) becomes \[ a_{7}={\frac {51}{33554432}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-{\frac {5}{8}}\) |
| \(a_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {7}{128}\) |
| \(a_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(\frac {3}{1024}\) |
| \(a_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(\frac {11}{32768}\) |
| \(a_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(\frac {13}{262144}\) |
| \(a_{6}\) | \(\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(\frac {35}{4194304}\) |
| \(a_{7}\) | \(\frac {\left (r^{2}+12 r +32\right ) \left (r^{2}-4\right ) r \left (-1+r \right ) \left (r +3\right )}{128 \left (2 r +13\right ) \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(\frac {51}{33554432}\) |
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 {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+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} -b_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+4 b_{n} \left (n +r \right ) \left (n +r -1\right )-b_{n -1} \left (n +r -1\right )+2 \left (n +r \right ) b_{n}+4 b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from recursive equation (4) gives \[ b_{n} = \frac {b_{n -1} \left (n^{2}+2 n r +r^{2}-2 n -2 r -3\right )}{4 n^{2}+8 n r +4 r^{2}-2 n -2 r}\tag {4} \] Which for the root \(r = 0\) becomes \[ b_{n} = \frac {b_{n -1} \left (n^{2}-2 n -3\right )}{4 n^{2}-2 n}\tag {5} \] 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 \[ b_{1}=\frac {r^{2}-4}{4 r^{2}+6 r +2} \] Which for the root \(r = 0\) becomes \[ b_{1}=-2 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120} \] Which for the root \(r = 0\) becomes \[ b_{3}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680} \] Which for the root \(r = 0\) becomes \[ b_{4}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240} \] Which for the root \(r = 0\) becomes \[ b_{5}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(0\) |
| \(b_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(0\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )} \] Which for the root \(r = 0\) becomes \[ b_{6}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(0\) |
| \(b_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(0\) |
| \(b_{6}\) | \(\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(0\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=\frac {\left (r^{2}+12 r +32\right ) \left (r^{2}-4\right ) r \left (-1+r \right ) \left (r +3\right )}{128 \left (2 r +13\right ) \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )} \] Which for the root \(r = 0\) becomes \[ b_{7}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r^{2}-4}{4 r^{2}+6 r +2}\) | \(-2\) |
| \(b_{2}\) | \(\frac {r^{3}-7 r +6}{16 r^{3}+48 r^{2}+44 r +12}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {\left (r +4\right ) r \left (-1+r \right ) \left (r -2\right )}{64 r^{4}+352 r^{3}+656 r^{2}+488 r +120}\) | \(0\) |
| \(b_{4}\) | \(\frac {\left (r -2\right ) \left (-1+r \right ) r \left (r +5\right )}{256 r^{4}+2048 r^{3}+5504 r^{2}+5632 r +1680}\) | \(0\) |
| \(b_{5}\) | \(\frac {\left (r +6\right ) \left (r +2\right ) r \left (r -2\right ) \left (-1+r \right )}{1024 r^{5}+12800 r^{4}+58880 r^{3}+121600 r^{2}+108096 r +30240}\) | \(0\) |
| \(b_{6}\) | \(\frac {\left (r^{2}+10 r +21\right ) \left (-1+r \right ) r \left (r^{2}-4\right )}{64 \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(0\) |
| \(b_{7}\) | \(\frac {\left (r^{2}+12 r +32\right ) \left (r^{2}-4\right ) r \left (-1+r \right ) \left (r +3\right )}{128 \left (2 r +13\right ) \left (2 r +11\right ) \left (32 r^{5}+400 r^{4}+1840 r^{3}+3800 r^{2}+3378 r +945\right )}\) | \(0\) |
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 \\ &= 1-2 x +\frac {x^{2}}{2}+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 {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+O\left (x^{8}\right )\right ) + c_{2} \left (1-2 x +\frac {x^{2}}{2}+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 {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {x^{2}}{2}+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 {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {x^{2}}{2}+O\left (x^{8}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \left (1-\frac {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {x^{2}}{2}+O\left (x^{8}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (\frac {d}{d x}y^{\prime }\right ) x \left (-4+x \right )+\left (2-x \right ) y^{\prime }+4 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 {4 y}{x \left (-4+x \right )}-\frac {\left (x -2\right ) y^{\prime }}{x \left (-4+x \right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (x -2\right ) y^{\prime }}{x \left (-4+x \right )}-\frac {4 y}{x \left (-4+x \right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {x -2}{x \left (-4+x \right )}, P_{3}\left (x \right )=-\frac {4}{x \left (-4+x \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}}}=\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}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x \left (-4+x \right )+\left (x -2\right ) y^{\prime }-4 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^{\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^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\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 -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\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} \\ {} & {} & -2 a_{0} r \left (-1+2 r \right ) x^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (-2 a_{k +1} \left (k +1+r \right ) \left (2 k +1+2 r \right )+a_{k} \left (k +r +2\right ) \left (k +r -2\right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -2 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 in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -4 \left (k +r +\frac {1}{2}\right ) \left (k +1+r \right ) a_{k +1}+a_{k} \left (k +r +2\right ) \left (k +r -2\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +r +2\right ) \left (k +r -2\right )}{2 \left (2 k +1+2 r \right ) \left (k +1+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0\hspace {3pt}\textrm {; series terminates at}\hspace {3pt} k =2 \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +2\right ) \left (k -2\right )}{2 \left (2 k +1\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =0 \\ {} & {} & a_{1}=-2 a_{0} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =1 \\ {} & {} & a_{2}=-\frac {a_{1}}{4} \\ \bullet & {} & \textrm {Express in terms of}\hspace {3pt} a_{0} \\ {} & {} & a_{2}=\frac {a_{0}}{2} \\ \bullet & {} & \textrm {Terminating series solution of the ODE for}\hspace {3pt} r =0\hspace {3pt}\textrm {. Use reduction of order to find the second linearly independent solution}\hspace {3pt} \\ {} & {} & y=a_{0}\cdot \left (1-2 x +\frac {1}{2} x^{2}\right ) \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +\frac {5}{2}\right ) \left (k -\frac {3}{2}\right )}{2 \left (2 k +2\right ) \left (k +\frac {3}{2}\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 +1}=\frac {a_{k} \left (k +\frac {5}{2}\right ) \left (k -\frac {3}{2}\right )}{2 \left (2 k +2\right ) \left (k +\frac {3}{2}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=a_{0}\cdot \left (1-2 x +\frac {1}{2} x^{2}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {1}{2}}\right ), b_{k +1}=\frac {b_{k} \left (k +\frac {5}{2}\right ) \left (k -\frac {3}{2}\right )}{2 \left (2 k +2\right ) \left (k +\frac {3}{2}\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)] Solution is available but has compositions of trig with ln functions of radicals. Attempting a simpler solution -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Reducible group (found another exponential solution) <- Kovacics algorithm successful <- linear_1 successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 42
Order:=8; dsolve(x*(4-x)*diff(y(x),x$2)+(2-x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {5}{8} x +\frac {7}{128} x^{2}+\frac {3}{1024} x^{3}+\frac {11}{32768} x^{4}+\frac {13}{262144} x^{5}+\frac {35}{4194304} x^{6}+\frac {51}{33554432} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 76
AsymptoticDSolveValue[x*(4-x)*y''[x]+(2-x)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^2}{2}-2 x+1\right )+c_1 \sqrt {x} \left (\frac {51 x^7}{33554432}+\frac {35 x^6}{4194304}+\frac {13 x^5}{262144}+\frac {11 x^4}{32768}+\frac {3 x^3}{1024}+\frac {7 x^2}{128}-\frac {5 x}{8}+1\right ) \]