30.13.12 problem 13

Internal problem ID [7644]
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 : 13
Date solved : Tuesday, September 30, 2025 at 04:55:25 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

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

\[ z = \left (1+\frac {x^{3}}{3}\right ) z \left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 15

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

\[ z(x)\to c_1 \left (\frac {x^3}{3}+1\right ) \]

✓ Sympy. Time used: 0.148 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
z = Function("z") 
ode = Eq(-x**2*z(x) + Derivative(z(x), x),0) 
ics = {} 
dsolve(ode,func=z(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

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