68.1.13 problem Problem 1.11(a)

Internal problem ID [14063]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.11(a)
Date solved : Wednesday, March 05, 2025 at 10:28:01 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 222

ode=x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} \sqrt [4]{x} \left (\frac {418854310875 i x^{9/2}}{8796093022208}-\frac {57972915 i x^{7/2}}{4294967296}+\frac {59535 i x^{5/2}}{8388608}-\frac {75 i x^{3/2}}{8192}-\frac {30241281245175 x^5}{281474976710656}+\frac {13043905875 x^4}{549755813888}-\frac {2401245 x^3}{268435456}+\frac {3675 x^2}{524288}-\frac {9 x}{512}+\frac {i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} \sqrt [4]{x} \left (-\frac {418854310875 i x^{9/2}}{8796093022208}+\frac {57972915 i x^{7/2}}{4294967296}-\frac {59535 i x^{5/2}}{8388608}+\frac {75 i x^{3/2}}{8192}-\frac {30241281245175 x^5}{281474976710656}+\frac {13043905875 x^4}{549755813888}-\frac {2401245 x^3}{268435456}+\frac {3675 x^2}{524288}-\frac {9 x}{512}-\frac {i \sqrt {x}}{16}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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