30.13.16 problem 17

Internal problem ID [7648]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 04:55:27 PM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 49

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

\[ w = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{20} x^{5}\right ) w \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) w^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ w(x)\to c_2 \left (\frac {x^5}{120}+\frac {x^4}{12}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^5}{20}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.208 (sec). Leaf size: 34

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

\[ w{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{12} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]