39.6.2 problem Problem 27.30

Internal problem ID [6560]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 27. Power series solutions of linear DE with variable coefficients. Supplementary Problems. page 274
Problem number : Problem 27.30
Date solved : Sunday, March 30, 2025 at 11:07:23 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{3} y^{\prime \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)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 222

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

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {468131288625 i x^{9/2}}{8796093022208}+\frac {66891825 i x^{7/2}}{4294967296}-\frac {72765 i x^{5/2}}{8388608}+\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {72765 i x^{5/2}}{8388608}-\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + 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)) + y(x) does not match hint 2nd_power_series_regular