Internal problem ID [6196]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) 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: 81
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+(1-2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\frac {1}{33177600} x^{6}+\frac {1}{3251404800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}-\frac {49}{331776000} x^{6}-\frac {121}{75866112000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 174
AsymptoticDSolveValue[4*x^2*y''[x]+(1-2*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {121 x^7}{75866112000}-\frac {49 x^6}{331776000}-\frac {137 x^5}{13824000}-\frac {25 x^4}{55296}-\frac {11 x^3}{864}-\frac {3 x^2}{16}-x\right )+\sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \log (x)\right ) \]