50.19.10 problem 3(a)

Internal problem ID [8118]
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 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 3(a)
Date solved : Sunday, March 30, 2025 at 12:45:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 37

Order:=8; 
ode:=x^3*diff(diff(y(x),x),x)+(-1+cos(2*x))*diff(y(x),x)+2*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {2}{9} x^{2}+\frac {26}{675} x^{4}-\frac {1742}{297675} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 x \left (1-\frac {1}{3} x^{2}+\frac {17}{270} x^{4}-\frac {173}{17010} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 74

ode=x^3*D[y[x],{x,2}]+(Cos[2*x]-1)*D[y[x],x]+2*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {32351 x^8}{40186125}-\frac {1742 x^6}{297675}+\frac {26 x^4}{675}-\frac {2 x^2}{9}+1\right ) x^2+c_1 \left (\frac {10471 x^8}{7144200}-\frac {173 x^6}{17010}+\frac {17 x^4}{270}-\frac {x^2}{3}+1\right ) x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + 2*x*y(x) + (cos(2*x) - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE x**3*Derivative(y(x), (x, 2)) + 2*x*y(x) + (cos(2*x) - 1)*Derivative(y(x), x) does not match hint 2nd_power_series_regular