50.23.3 problem 1(c)

Internal problem ID [8166]
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)
Date solved : Monday, January 27, 2025 at 03:45:07 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+p \left (p +1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} \infty \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 246

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);
 

\[ y = \frac {c_{1} \left (\frac {1}{x}\right )^{p} \left (1+\frac {p^{2}+3 p +2}{2 \left (2 p +3\right ) x^{2}}+\frac {p^{4}+10 p^{3}+35 p^{2}+50 p +24}{8 \left (2 p +5\right ) \left (2 p +3\right ) x^{4}}+\frac {p^{6}+21 p^{5}+175 p^{4}+735 p^{3}+1624 p^{2}+1764 p +720}{48 \left (2 p +7\right ) \left (2 p +5\right ) \left (2 p +3\right ) x^{6}}+O\left (\frac {1}{x^{8}}\right )\right )}{x}+c_{2} \left (\frac {1}{x}\right )^{-p} \left (1-\frac {p \left (p -1\right )}{2 \left (2 p -1\right ) x^{2}}+\frac {p \left (p^{3}-6 p^{2}+11 p -6\right )}{8 \left (2 p -3\right ) \left (2 p -1\right ) x^{4}}-\frac {p \left (p^{5}-15 p^{4}+85 p^{3}-225 p^{2}+274 p -120\right )}{48 \left (2 p -5\right ) \left (2 p -3\right ) \left (2 p -1\right ) x^{6}}+O\left (\frac {1}{x^{8}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 2707

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+p*(p+1)*y[x]==0,y[x],{x,Infinity,"8"-1}]
 

\[ 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 \]