40.16.9 problem 16

Internal problem ID [6800]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 25. Integration in series. Supplemetary problems. Page 205
Problem number : 16
Date solved : Monday, January 27, 2025 at 02:30:45 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*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 81

Order:=6; 
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=0);
 

\[ y = \left (1-\frac {p \left (p +1\right ) x^{2}}{2}+\frac {p \left (p^{3}+2 p^{2}-5 p -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}+p -2\right ) x^{3}}{6}+\frac {\left (p^{4}+2 p^{3}-13 p^{2}-14 p +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 120

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+p*(p+1)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {1}{120} \left (p^2+p\right )^2 x^5+\frac {7}{60} \left (-p^2-p\right ) x^5+\frac {1}{6} \left (-p^2-p\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (p^2+p\right )^2 x^4+\frac {1}{4} \left (-p^2-p\right ) x^4+\frac {1}{2} \left (-p^2-p\right ) x^2+1\right ) \]