Internal problem ID [6268]
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: 20.
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 }+2 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 39
Order:=8; dsolve(x*diff(y(x),x$2)+(1-x^2)*diff(y(x),x)+2*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\frac {3}{4} x^{2}-\frac {1}{32} x^{4}-\frac {1}{576} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (1-\frac {1}{2} x^{2}+\mathrm {O}\left (x^{8}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 53
AsymptoticDSolveValue[x*y''[x]+(1-x^2)*y'[x]+2*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (1-\frac {x^2}{2}\right )+c_2 \left (-\frac {x^6}{576}-\frac {x^4}{32}+\frac {3 x^2}{4}+\left (1-\frac {x^2}{2}\right ) \log (x)\right ) \]