Internal problem ID [6243]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page
391
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (x +2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 52
Order:=8; dsolve(2*x*(1-x)*diff(y(x),x$2)+(1-2*x)*diff(y(x),x)+(2+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{2} x -\frac {9}{40} x^{2}-\frac {149}{1680} x^{3}-\frac {661}{13440} x^{4}-\frac {16171}{492800} x^{5}-\frac {5530601}{230630400} x^{6}-\frac {299137703}{16144128000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x -\frac {1}{6} x^{2}+\frac {1}{15} x^{3}+\frac {37}{840} x^{4}+\frac {527}{18900} x^{5}+\frac {16309}{831600} x^{6}+\frac {14339}{970200} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 111
AsymptoticDSolveValue[2*x*(1-x)*y''[x]+(1-2*x)*y'[x]+(2+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {299137703 x^7}{16144128000}-\frac {5530601 x^6}{230630400}-\frac {16171 x^5}{492800}-\frac {661 x^4}{13440}-\frac {149 x^3}{1680}-\frac {9 x^2}{40}-\frac {x}{2}+1\right )+c_2 \left (\frac {14339 x^7}{970200}+\frac {16309 x^6}{831600}+\frac {527 x^5}{18900}+\frac {37 x^4}{840}+\frac {x^3}{15}-\frac {x^2}{6}-2 x+1\right ) \]