88.24.11 problem 11

Internal problem ID [24204]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 202
Problem number : 11
Date solved : Friday, October 03, 2025 at 08:03:53 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 43

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

\[ y = \left (1+\frac {x^{3}}{3}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+\frac {x^{2} y^{\prime \prime }\left (0\right )}{2}+\frac {y^{\prime \prime }\left (0\right ) x^{5}}{30}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to c_2 \left (\frac {x^4}{12}+x\right )+c_1 \left (\frac {x^3}{3}+1\right )+c_3 \left (\frac {x^5}{30}+\frac {x^2}{2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*y(x) + (x**3 - 8)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

Series solution not supported for ode of order > 2