Internal problem ID [5275]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 3. Linear equations with variable coefficients. Page 130
Problem number: 6.
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 }+\alpha \left (\alpha +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 101
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+alpha*(alpha+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {\alpha \left (\alpha +1\right ) x^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) x^{4}}{24}\right ) y \relax (0)+\left (x -\frac {\left (\alpha ^{2}+\alpha -2\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 127
AsymptoticDSolveValue[(1-x^2)*y''[x]-2*x*y'[x]+\[Alpha]*(\[Alpha]+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {1}{60} \left (-\alpha ^2-\alpha \right ) x^5-\frac {1}{120} \left (-\alpha ^2-\alpha \right ) \left (\alpha ^2+\alpha \right ) x^5-\frac {1}{10} \left (\alpha ^2+\alpha \right ) x^5+\frac {x^5}{5}-\frac {1}{6} \left (\alpha ^2+\alpha \right ) x^3+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (\alpha ^2+\alpha \right )^2 x^4-\frac {1}{4} \left (\alpha ^2+\alpha \right ) x^4-\frac {1}{2} \left (\alpha ^2+\alpha \right ) x^2+1\right ) \]