Internal problem ID [6149]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page
355
Problem number: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }-4 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: 34
Order:=8; dsolve((1-4*x^2)*diff(y(x),x$2)+6*x*diff(y(x),x)-4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (2 x^{2}+1\right ) y \relax (0)+\left (x -\frac {1}{3} x^{3}-\frac {1}{6} x^{5}-\frac {3}{14} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 40
AsymptoticDSolveValue[(1-4*x^2)*y''[x]+6*x*y'[x]-4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (2 x^2+1\right )+c_2 \left (-\frac {3 x^7}{14}-\frac {x^5}{6}-\frac {x^3}{3}+x\right ) \]