87.26.34 problem 46

Internal problem ID [23868]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 46
Date solved : Thursday, October 02, 2025 at 09:46:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-\frac {1}{4}\right ) y^{\prime \prime }+2 y^{\prime }-6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 59

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

\[ y = \left (1-12 x^{2}-32 x^{3}-48 x^{4}-\frac {384}{5} x^{5}\right ) y \left (0\right )+\left (x +4 x^{2}+\frac {20}{3} x^{3}+8 x^{4}+\frac {64}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 60

ode=(x^2-1/4)*D[y[x],{x,2}]+2*D[y[x],x]-6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {384 x^5}{5}-48 x^4-32 x^3-12 x^2+1\right )+c_2 \left (\frac {64 x^5}{5}+8 x^4+\frac {20 x^3}{3}+4 x^2+x\right ) \]

✓ Sympy. Time used: 0.317 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (- 48 x^{4} - 32 x^{3} - 12 x^{2} + 1\right ) + C_{1} x \left (8 x^{3} + \frac {20 x^{2}}{3} + 4 x + 1\right ) + O\left (x^{6}\right ) \]