88.25.2 problem 2 (b)

Internal problem ID [24207]
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 212
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 10:00:49 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {2}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 39

ode=x*D[y[x],{x,2}]+D[y[x],{x,1}]-x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{9}+1\right )+c_2 \left (\left (\frac {x^3}{9}+1\right ) \log (x)-\frac {2 x^3}{27}\right ) \]

✓ Sympy. Time used: 0.207 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{3}}{9} + 1\right ) + O\left (x^{6}\right ) \]