Internal problem ID [11904]
Internal file name [OUTPUT/11914_Saturday_April_13_2024_10_26_15_PM_30461767/index.tex]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 4.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+y^{\prime } x^{2}+y \left (x -2\right )=0} \] 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. \[ \left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+y^{\prime } x^{2}+y \left (x -2\right ) = 0 \] 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 {1}{x \left (x^{2}+x -6\right )}\\ q(x) &= \frac {1}{\left (x +3\right ) x^{3}}\\ \end {align*}
| \(p(x)=\frac {1}{x \left (x^{2}+x -6\right )}\) | |
| singularity | type |
| \(x = -3\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 2\) | \(\text {``regular''}\) |
| \(q(x)=\frac {1}{\left (x +3\right ) x^{3}}\) | |
| singularity | type |
| \(x = -3\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``irregular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-3, 2, \infty ]\)
Irregular singular points : \([0]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+y^{\prime } x^{2}+y \left (x -2\right )=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 {y}{\left (x +3\right ) x^{3}}-\frac {y^{\prime }}{x \left (x^{2}+x -6\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 {y^{\prime }}{x \left (x^{2}+x -6\right )}+\frac {y}{\left (x +3\right ) x^{3}}=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 {1}{x \left (x^{2}+x -6\right )}, P_{3}\left (x \right )=\frac {1}{\left (x +3\right ) x^{3}}\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}}}=\frac {1}{15} \\ {} & \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} \\ {} & {} & y^{\prime \prime } x^{3} \left (x^{2}+x -6\right ) \left (x +3\right )+y^{\prime } x^{2} \left (x +3\right )+y \left (x^{2}+x -6\right )=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 (u^{6}-14 u^{5}+72 u^{4}-162 u^{3}+135 u^{2}\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (u^{3}-6 u^{2}+9 u \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (u^{2}-5 u \right ) 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 y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \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 =1..3 \\ {} & {} & 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 =2..6 \\ {} & {} & 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} \\ {} & {} & 9 a_{0} r \left (-14+15 r \right ) u^{r}+\left (9 a_{1} \left (1+r \right ) \left (1+15 r \right )-a_{0} \left (162 r^{2}-156 r +5\right )\right ) u^{1+r}+\left (9 a_{2} \left (2+r \right ) \left (16+15 r \right )-a_{1} \left (162 r^{2}+168 r +11\right )+a_{0} \left (72 r^{2}-71 r +1\right )\right ) u^{2+r}+\left (9 a_{3} \left (3+r \right ) \left (31+15 r \right )-a_{2} \left (162 r^{2}+492 r +341\right )+a_{1} \left (72 r^{2}+73 r +2\right )-14 a_{0} r \left (-1+r \right )\right ) u^{3+r}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (9 a_{k} \left (k +r \right ) \left (15 k +15 r -14\right )-a_{k -1} \left (162 \left (k -1\right )^{2}+324 \left (k -1\right ) r +162 r^{2}-156 k +161-156 r \right )+a_{k -2} \left (72 \left (k -2\right )^{2}+144 \left (k -2\right ) r +72 r^{2}-71 k +143-71 r \right )-14 a_{k -3} \left (k -3+r \right ) \left (k -4+r \right )+a_{k -4} \left (k -4+r \right ) \left (k -5+r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 9 r \left (-14+15 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {14}{15}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [9 a_{1} \left (1+r \right ) \left (1+15 r \right )-a_{0} \left (162 r^{2}-156 r +5\right )=0, 9 a_{2} \left (2+r \right ) \left (16+15 r \right )-a_{1} \left (162 r^{2}+168 r +11\right )+a_{0} \left (72 r^{2}-71 r +1\right )=0, 9 a_{3} \left (3+r \right ) \left (31+15 r \right )-a_{2} \left (162 r^{2}+492 r +341\right )+a_{1} \left (72 r^{2}+73 r +2\right )-14 a_{0} r \left (-1+r \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (162 r^{2}-156 r +5\right )}{9 \left (15 r^{2}+16 r +1\right )}, a_{2}=\frac {a_{0} \left (16524 r^{4}+1161 r^{3}-14175 r^{2}-381 r +46\right )}{81 \left (225 r^{4}+930 r^{3}+1231 r^{2}+558 r +32\right )}, a_{3}=\frac {a_{0} \left (1357398 r^{6}+4208274 r^{5}+2090772 r^{4}-3149874 r^{3}-2550699 r^{2}-162981 r +12806\right )}{729 \left (3375 r^{6}+31050 r^{5}+110070 r^{4}+188416 r^{3}+157371 r^{2}+54326 r +2976\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (135 a_{k}+a_{k -4}-14 a_{k -3}+72 a_{k -2}-162 a_{k -1}\right ) k^{2}+\left (2 \left (135 a_{k}+a_{k -4}-14 a_{k -3}+72 a_{k -2}-162 a_{k -1}\right ) r -126 a_{k}-9 a_{k -4}+98 a_{k -3}-359 a_{k -2}+480 a_{k -1}\right ) k +\left (135 a_{k}+a_{k -4}-14 a_{k -3}+72 a_{k -2}-162 a_{k -1}\right ) r^{2}+\left (-126 a_{k}-9 a_{k -4}+98 a_{k -3}-359 a_{k -2}+480 a_{k -1}\right ) r +20 a_{k -4}-168 a_{k -3}+431 a_{k -2}-323 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (135 a_{k +4}+a_{k}-14 a_{k +1}+72 a_{k +2}-162 a_{k +3}\right ) \left (k +4\right )^{2}+\left (2 \left (135 a_{k +4}+a_{k}-14 a_{k +1}+72 a_{k +2}-162 a_{k +3}\right ) r -126 a_{k +4}-9 a_{k}+98 a_{k +1}-359 a_{k +2}+480 a_{k +3}\right ) \left (k +4\right )+\left (135 a_{k +4}+a_{k}-14 a_{k +1}+72 a_{k +2}-162 a_{k +3}\right ) r^{2}+\left (-126 a_{k +4}-9 a_{k}+98 a_{k +1}-359 a_{k +2}+480 a_{k +3}\right ) r +20 a_{k}-168 a_{k +1}+431 a_{k +2}-323 a_{k +3}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}+2 k r a_{k}-28 k r a_{k +1}+144 k r a_{k +2}-324 k r a_{k +3}+r^{2} a_{k}-14 r^{2} a_{k +1}+72 r^{2} a_{k +2}-162 r^{2} a_{k +3}-k a_{k}-14 k a_{k +1}+217 k a_{k +2}-816 k a_{k +3}-r a_{k}-14 r a_{k +1}+217 r a_{k +2}-816 r a_{k +3}+147 a_{k +2}-995 a_{k +3}}{9 \left (15 k^{2}+30 k r +15 r^{2}+106 k +106 r +184\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}-k a_{k}-14 k a_{k +1}+217 k a_{k +2}-816 k a_{k +3}+147 a_{k +2}-995 a_{k +3}}{9 \left (15 k^{2}+106 k +184\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}-k a_{k}-14 k a_{k +1}+217 k a_{k +2}-816 k a_{k +3}+147 a_{k +2}-995 a_{k +3}}{9 \left (15 k^{2}+106 k +184\right )}, a_{1}=\frac {5 a_{0}}{9}, a_{2}=\frac {23 a_{0}}{1296}, a_{3}=\frac {6403 a_{0}}{1084752}\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}, a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}-k a_{k}-14 k a_{k +1}+217 k a_{k +2}-816 k a_{k +3}+147 a_{k +2}-995 a_{k +3}}{9 \left (15 k^{2}+106 k +184\right )}, a_{1}=\frac {5 a_{0}}{9}, a_{2}=\frac {23 a_{0}}{1296}, a_{3}=\frac {6403 a_{0}}{1084752}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {14}{15} \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}+\frac {13}{15} k a_{k}-\frac {602}{15} k a_{k +1}+\frac {1757}{5} k a_{k +2}-\frac {5592}{5} k a_{k +3}-\frac {14}{225} a_{k}-\frac {5684}{225} a_{k +1}+\frac {30919}{75} a_{k +2}-\frac {47443}{25} a_{k +3}}{9 \left (15 k^{2}+134 k +296\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {14}{15} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {14}{15}}, a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}+\frac {13}{15} k a_{k}-\frac {602}{15} k a_{k +1}+\frac {1757}{5} k a_{k +2}-\frac {5592}{5} k a_{k +3}-\frac {14}{225} a_{k}-\frac {5684}{225} a_{k +1}+\frac {30919}{75} a_{k +2}-\frac {47443}{25} a_{k +3}}{9 \left (15 k^{2}+134 k +296\right )}, a_{1}=\frac {13 a_{0}}{6525}, a_{2}=\frac {64478 a_{0}}{16149375}, a_{3}=\frac {1058799374 a_{0}}{643148859375}\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 +\frac {14}{15}}, a_{k +4}=-\frac {k^{2} a_{k}-14 k^{2} a_{k +1}+72 k^{2} a_{k +2}-162 k^{2} a_{k +3}+\frac {13}{15} k a_{k}-\frac {602}{15} k a_{k +1}+\frac {1757}{5} k a_{k +2}-\frac {5592}{5} k a_{k +3}-\frac {14}{225} a_{k}-\frac {5684}{225} a_{k +1}+\frac {30919}{75} a_{k +2}-\frac {47443}{25} a_{k +3}}{9 \left (15 k^{2}+134 k +296\right )}, a_{1}=\frac {13 a_{0}}{6525}, a_{2}=\frac {64478 a_{0}}{16149375}, a_{3}=\frac {1058799374 a_{0}}{643148859375}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +3\right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +3\right )^{k +\frac {14}{15}}\right ), a_{k +4}=-\frac {k^{2} a_{k}+72 k^{2} a_{2+k}-162 k^{2} a_{3+k}-14 k^{2} a_{k +1}-k a_{k}+217 k a_{2+k}-816 k a_{3+k}-14 k a_{k +1}+147 a_{2+k}-995 a_{3+k}}{9 \left (15 k^{2}+106 k +184\right )}, a_{1}=\frac {5 a_{0}}{9}, a_{2}=\frac {23 a_{0}}{1296}, a_{3}=\frac {6403 a_{0}}{1084752}, b_{k +4}=-\frac {k^{2} b_{k}-14 k^{2} b_{k +1}+72 k^{2} b_{2+k}-162 k^{2} b_{3+k}+\frac {13}{15} k b_{k}-\frac {602}{15} k b_{k +1}+\frac {1757}{5} k b_{2+k}-\frac {5592}{5} k b_{3+k}-\frac {14}{225} b_{k}-\frac {5684}{225} b_{k +1}+\frac {30919}{75} b_{2+k}-\frac {47443}{25} b_{3+k}}{9 \left (15 k^{2}+134 k +296\right )}, b_{1}=\frac {13 b_{0}}{6525}, b_{2}=\frac {64478 b_{0}}{16149375}, b_{3}=\frac {1058799374 b_{0}}{643148859375}\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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c <> 0 `
✗ Solution by Maple
Order:=6; dsolve((x^5+x^4-6*x^3)*diff(y(x),x$2)+x^2*diff(y(x),x)+(x-2)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.226 (sec). Leaf size: 282
AsymptoticDSolveValue[(x^5+x^4-6*x^3)*y''[x]+x^2*y'[x]+(x-2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (-\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}+\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}-\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}+\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}-\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right )+c_2 e^{\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}-\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}+\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}-\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}+\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right ) \]