Internal problem ID [6255]
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: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }-18 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: 44
Order:=8; dsolve((1-x^2)*diff(y(x),x$2)-10*x*diff(y(x),x)-18*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (70 x^{6}+30 x^{4}+9 x^{2}+1\right ) y \relax (0)+\left (x +\frac {14}{3} x^{3}+\frac {63}{5} x^{5}+\frac {132}{5} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 50
AsymptoticDSolveValue[(1-x^2)*y''[x]-10*x*y'[x]-18*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {132 x^7}{5}+\frac {63 x^5}{5}+\frac {14 x^3}{3}+x\right )+c_1 \left (70 x^6+30 x^4+9 x^2+1\right ) \]