Internal problem ID [6207]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
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 y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 45
Order:=8; dsolve(x*diff(y(x),x$2)+(1-x^2)*diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{4} x^{2}+\frac {3}{64} x^{4}+\frac {5}{768} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\left (-\frac {1}{128} x^{4}-\frac {1}{512} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 74
AsymptoticDSolveValue[x*y''[x]+(1-x^2)*y'[x]-x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {5 x^6}{768}+\frac {3 x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (-\frac {x^6}{512}-\frac {x^4}{128}+\left (\frac {5 x^6}{768}+\frac {3 x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]