67.25.13 problem 35.3 (g)

Internal problem ID [17008]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (g)
Date solved : Thursday, October 02, 2025 at 01:41:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \left (1+\frac {1}{6} x^{2}+\frac {1}{9} x^{3}+\frac {1}{24} x^{4}+\frac {31}{270} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +\ln \left (x \right ) \left (\left (-4\right ) x -\frac {2}{3} x^{3}-\frac {4}{9} x^{4}-\frac {1}{6} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+4 x -\frac {7}{2} x^{2}+\frac {14}{9} x^{3}+\frac {133}{216} x^{4}+\frac {23}{90} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 79

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

\[ y(x)\to c_2 \left (\frac {x^4}{24}+\frac {x^3}{9}+\frac {x^2}{6}+1\right )+c_1 \left (\frac {229 x^4+480 x^3-756 x^2+1728 x+216}{216 x}-\frac {2}{9} \left (2 x^3+3 x^2+18\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.622 (sec). Leaf size: 10

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

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