54.8.2 problem 2

Internal problem ID [8664]
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
Date solved : Monday, January 27, 2025 at 04:18:43 PM
CAS classification : [_Jacobi]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (x +2\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (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);
 

\[ y = c_{1} \sqrt {x}\, \left (1-\frac {1}{2} x -\frac {9}{40} x^{2}-\frac {149}{1680} x^{3}-\frac {661}{13440} x^{4}-\frac {16171}{492800} x^{5}-\frac {5530601}{230630400} x^{6}-\frac {299137703}{16144128000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x -\frac {1}{6} x^{2}+\frac {1}{15} x^{3}+\frac {37}{840} x^{4}+\frac {527}{18900} x^{5}+\frac {16309}{831600} x^{6}+\frac {14339}{970200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 111

AsymptoticDSolveValue[2*x*(1-x)*D[y[x],{x,2}]+(1-2*x)*D[y[x],x]+(2+x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {299137703 x^7}{16144128000}-\frac {5530601 x^6}{230630400}-\frac {16171 x^5}{492800}-\frac {661 x^4}{13440}-\frac {149 x^3}{1680}-\frac {9 x^2}{40}-\frac {x}{2}+1\right )+c_2 \left (\frac {14339 x^7}{970200}+\frac {16309 x^6}{831600}+\frac {527 x^5}{18900}+\frac {37 x^4}{840}+\frac {x^3}{15}-\frac {x^2}{6}-2 x+1\right ) \]