14.28.14 problem 20

Internal problem ID [4076]
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
Date solved : Tuesday, September 30, 2025 at 07:01:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (1-\frac {3}{4 x^{2}}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 46

Order:=6; 
ode:=diff(diff(y(x),x),x)+(1-3/4/x^2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 72

ode=D[y[x],{x,2}]+(1-3/(4*x^2))*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^{11/2}}{192}-\frac {x^{7/2}}{8}+x^{3/2}\right )+c_1 \left (\frac {1}{16} x^{3/2} \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.267 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 3/(4*x**2))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{\frac {3}{2}} \left (1 - \frac {x^{2}}{8}\right ) + O\left (x^{6}\right ) \]