Internal problem ID [6261]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-x \left (x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 51
Order:=8; dsolve(x^2*diff(y(x),x$2)-x*(1+x^2)*diff(y(x),x)+(1-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{32} x^{4}-\frac {11}{576} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 86
AsymptoticDSolveValue[x^2*y''[x]-x*(1+x^2)*y'[x]+(1-x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (x \left (-\frac {11 x^6}{576}-\frac {3 x^4}{32}-\frac {x^2}{4}\right )+x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \log (x)\right ) \]