Internal problem ID [5745]
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. Problems for review and discovert.
(B) Challenge Problems . Page 194
Problem number: 1(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+p \left (p +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = \infty \).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1124
Order:=8; dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+p*(p+1)*y(x)=0,y(x),type='series',x=Infinity);
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 2707
AsymptoticDSolveValue[(1-x^2)*y''[x]-2*x*y'[x]+p*(p+1)*y[x]==0,y[x],{x,Infinity,7}]
\[ y(x)\to \left (\frac {p^2 x^{-p-7}}{-p^2-p+(p+6) (p+7)}+\frac {3 p x^{-p-7}}{-p^2-p+(p+6) (p+7)}+\frac {p^4 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {6 p^3 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {17 p^2 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {24 p x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {12 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {p^4 x^{-p-7}}{\left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {6 p^3 x^{-p-7}}{\left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {21 p^2 x^{-p-7}}{\left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {36 p x^{-p-7}}{\left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {p^6 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {9 p^5 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {45 p^4 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {135 p^3 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {254 p^2 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {276 p x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {120 x^{-p-7}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {20 x^{-p-7}}{\left (-p^2-p+(p+4) (p+5)\right ) \left (-p^2-p+(p+6) (p+7)\right )}+\frac {2 x^{-p-7}}{-p^2-p+(p+6) (p+7)}+\frac {p^2 x^{-p-5}}{-p^2-p+(p+4) (p+5)}+\frac {3 p x^{-p-5}}{-p^2-p+(p+4) (p+5)}+\frac {p^4 x^{-p-5}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right )}+\frac {6 p^3 x^{-p-5}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right )}+\frac {17 p^2 x^{-p-5}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right )}+\frac {24 p x^{-p-5}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right )}+\frac {12 x^{-p-5}}{\left (-p^2-p+(p+2) (p+3)\right ) \left (-p^2-p+(p+4) (p+5)\right )}+\frac {2 x^{-p-5}}{-p^2-p+(p+4) (p+5)}+\frac {p^2 x^{-p-3}}{-p^2-p+(p+2) (p+3)}+\frac {3 p x^{-p-3}}{-p^2-p+(p+2) (p+3)}+\frac {2 x^{-p-3}}{-p^2-p+(p+2) (p+3)}+x^{-p-1}\right ) c_1+\left (\frac {p^2 x^{p-6}}{-p^2-p+(5-p) (6-p)}-\frac {p x^{p-6}}{-p^2-p+(5-p) (6-p)}+\frac {p^4 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {2 p^3 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {5 p^2 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {4 p x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {p^4 x^{p-6}}{\left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {2 p^3 x^{p-6}}{\left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {9 p^2 x^{p-6}}{\left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {8 p x^{p-6}}{\left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {p^6 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {3 p^5 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {15 p^4 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {25 p^3 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {44 p^2 x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}-\frac {32 p x^{p-6}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right ) \left (-p^2-p+(5-p) (6-p)\right )}+\frac {p^2 x^{p-4}}{-p^2-p+(3-p) (4-p)}-\frac {p x^{p-4}}{-p^2-p+(3-p) (4-p)}+\frac {p^4 x^{p-4}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right )}-\frac {2 p^3 x^{p-4}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right )}+\frac {5 p^2 x^{p-4}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right )}-\frac {4 p x^{p-4}}{\left (-p^2-p+(1-p) (2-p)\right ) \left (-p^2-p+(3-p) (4-p)\right )}+\frac {p^2 x^{p-2}}{-p^2-p+(1-p) (2-p)}-\frac {p x^{p-2}}{-p^2-p+(1-p) (2-p)}+x^p\right ) c_2 \]