50.20.2 problem 2

Internal problem ID [8130]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:46:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-8 x^{2} y^{\prime }+\left (4 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 52

Order:=8; 
ode:=4*x^2*diff(diff(y(x),x),x)-8*x^2*diff(y(x),x)+(4*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) \left (c_1 +c_2 \ln \left (x \right )\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 112

ode=4*x^2*D[y[x],{x,2}]-8*x^2*D[y[x],x]+(4*x^2+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \sqrt {x} \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \log (x) \]

✓ Sympy. Time used: 0.882 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**2*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + (4*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {x^{6}}{720} + \frac {x^{5}}{120} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + O\left (x^{8}\right ) \]