Internal problem ID [4212]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_elliptic, _class_II]]
Solve \begin {gather*} \boxed {x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
Order:=6; dsolve(x*(1-x^2)*diff(y(x),x$2)+(1-x^2)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} x^{2}-\frac {3}{64} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {1}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 60
AsymptoticDSolveValue[x*(1-x^2)*y''[x]+(1-x^2)*y'[x]+x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {x^4}{128}+\frac {x^2}{4}+\left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]