Internal problem ID [4202]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+a^{2} 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: 71
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+a^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {a^{2} x^{2}}{2}+\frac {a^{2} \left (a^{2}-4\right ) x^{4}}{24}\right ) y \relax (0)+\left (x -\frac {\left (a^{2}-1\right ) x^{3}}{6}+\frac {\left (a^{4}-10 a^{2}+9\right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 88
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+a^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {a^4 x^5}{120}-\frac {a^2 x^5}{12}-\frac {a^2 x^3}{6}+\frac {3 x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {a^4 x^4}{24}-\frac {a^2 x^4}{6}-\frac {a^2 x^2}{2}+1\right ) \]