Internal problem ID [5721]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.6. Gauss’s Hypergeometric
Equation. Page 187
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+p^{2} y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 338
Order:=8; dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+p^2*y(x)=0,y(x),type='series',x=1);
\[ y \relax (x ) = c_{1} \sqrt {x -1}\, \left (1+\left (\frac {p^{2}}{3}-\frac {1}{12}\right ) \left (x -1\right )+\left (\frac {1}{30} p^{4}-\frac {1}{12} p^{2}+\frac {3}{160}\right ) \left (x -1\right )^{2}+\left (\frac {1}{630} p^{6}-\frac {1}{72} p^{4}+\frac {37}{1440} p^{2}-\frac {5}{896}\right ) \left (x -1\right )^{3}+\left (\frac {1}{22680} p^{8}-\frac {1}{1080} p^{6}+\frac {47}{8640} p^{4}-\frac {3229}{362880} p^{2}+\frac {35}{18432}\right ) \left (x -1\right )^{4}+\left (\frac {1}{1247400} p^{10}-\frac {1}{30240} p^{8}+\frac {19}{43200} p^{6}-\frac {1571}{725760} p^{4}+\frac {10679}{3225600} p^{2}-\frac {63}{90112}\right ) \left (x -1\right )^{5}+\left (\frac {1}{97297200} p^{12}-\frac {1}{1360800} p^{10}+\frac {67}{3628800} p^{8}-\frac {2159}{10886400} p^{6}+\frac {153617}{174182400} p^{4}-\frac {550499}{425779200} p^{2}+\frac {231}{851968}\right ) \left (x -1\right )^{6}+\left (\frac {1}{10216206000} p^{14}-\frac {1}{89812800} p^{12}+\frac {11}{23328000} p^{10}-\frac {8521}{914457600} p^{8}+\frac {230443}{2612736000} p^{6}-\frac {1206053}{3284582400} p^{4}+\frac {2430898831}{4649508864000} p^{2}-\frac {143}{1310720}\right ) \left (x -1\right )^{7}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right )+c_{2} \left (1+p^{2} \left (x -1\right )+\left (\frac {1}{6} p^{4}-\frac {1}{6} p^{2}\right ) \left (x -1\right )^{2}+\left (\frac {1}{90} p^{6}-\frac {1}{18} p^{4}+\frac {2}{45} p^{2}\right ) \left (x -1\right )^{3}+\left (\frac {1}{2520} p^{8}-\frac {1}{180} p^{6}+\frac {7}{360} p^{4}-\frac {1}{70} p^{2}\right ) \left (x -1\right )^{4}+\left (\frac {1}{113400} p^{10}-\frac {1}{3780} p^{8}+\frac {13}{5400} p^{6}-\frac {41}{5670} p^{4}+\frac {8}{1575} p^{2}\right ) \left (x -1\right )^{5}+\left (\frac {1}{7484400} p^{12}-\frac {1}{136080} p^{10}+\frac {31}{226800} p^{8}-\frac {139}{136080} p^{6}+\frac {479}{170100} p^{4}-\frac {4}{2079} p^{2}\right ) \left (x -1\right )^{6}+\left (\frac {1}{681080400} p^{14}-\frac {1}{7484400} p^{12}+\frac {1}{226800} p^{10}-\frac {311}{4762800} p^{8}+\frac {37}{85050} p^{6}-\frac {59}{51975} p^{4}+\frac {16}{21021} p^{2}\right ) \left (x -1\right )^{7}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 5699
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+p^2*y[x]==0,y[x],{x,1,7}]
Too large to display