78.19.9 problem 2 (e)

Internal problem ID [18349]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 2 (e)
Date solved : Thursday, March 13, 2025 at 11:54:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 222

ode=x^4*D[y[x],{x,2}]+Sin[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {12579783586699513 i x^{9/2}}{96185277197844480}-\frac {21896783401 i x^{7/2}}{579820584960}+\frac {856783 i x^{5/2}}{41943040}-\frac {3151 i x^{3/2}}{73728}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {12579783586699513 i x^{9/2}}{96185277197844480}+\frac {21896783401 i x^{7/2}}{579820584960}-\frac {856783 i x^{5/2}}{41943040}+\frac {3151 i x^{3/2}}{73728}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \]

✗ Sympy

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

ValueError : ODE x**4*Derivative(y(x), (x, 2)) + y(x)*sin(x) does not match hint 2nd_power_series_regular