88.25.15 problem 13

Internal problem ID [24220]
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 : 13
Date solved : Thursday, October 02, 2025 at 10:00:56 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }+2 y x&=x^{3}-x +3 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 49

ode=D[y[x],{x,2}]+x^2*D[y[x],{x,1}]+2*x*y[x]==3-x+x^3; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {x^5}{4}+c_2 \left (x-\frac {x^4}{4}\right )-\frac {x^3}{6}+c_1 \left (1-\frac {x^3}{3}\right )+\frac {3 x^2}{2} \]

✗ Sympy

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

ValueError : ODE -x**3 + x**2*Derivative(y(x), x) + 2*x*y(x) + x + Derivative(y(x), (x, 2)) - 3 does