Internal problem ID [5856]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 39
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1/4)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\mathrm {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 76
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-1/4)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^{11/2}}{720}+\frac {x^{7/2}}{24}-\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (-\frac {x^{13/2}}{5040}+\frac {x^{9/2}}{120}-\frac {x^{5/2}}{6}+\sqrt {x}\right ) \]