Internal problem ID [5457]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +p \left (p +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 101
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)*y''[x]-2*x*y'[x]+p*(p+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {1}{120} \left (p^2+p\right )^2 x^5+\frac {7}{60} \left (-p^2-p\right ) x^5+\frac {1}{6} \left (-p^2-p\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (p^2+p\right )^2 x^4+\frac {1}{4} \left (-p^2-p\right ) x^4+\frac {1}{2} \left (-p^2-p\right ) x^2+1\right ) \]