88.24.9 problem 9

Internal problem ID [24202]
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 : 9
Date solved : Thursday, October 02, 2025 at 10:00:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+8\right ) y^{\prime \prime }+3 x^{2} y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

Order:=6; 
ode:=(x^3+8)*diff(diff(y(x),x),x)+3*x^2*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{3}}{48}\right ) y \left (0\right )+\left (x -\frac {1}{24} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=(8+x^3)*D[y[x],{x,2}]+3*x^2*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x-\frac {x^4}{24}\right )+c_1 \left (1-\frac {x^3}{48}\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), x) + 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)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{3}}{48}\right ) + C_{1} x \left (1 - \frac {x^{3}}{24}\right ) + O\left (x^{6}\right ) \]