Internal problem ID [2478]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Additional problems. Section
11.7. page 788
Problem number: 20.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (1-\frac {3}{4 x^{2}}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 49
Order:=6; dsolve(diff(y(x),x$2)+(1-3/(4*x^2))*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (x^{2}-\frac {1}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 72
AsymptoticDSolveValue[y''[x]+(1-3/(4*x^2))*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{11/2}}{192}-\frac {x^{7/2}}{8}+x^{3/2}\right )+c_1 \left (\frac {1}{16} x^{3/2} \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 \sqrt {x}}\right ) \]