82.1.6 problem 23-10

Internal problem ID [21748]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 23. Power series. Page 695
Problem number : 23-10
Date solved : Thursday, October 02, 2025 at 08:01:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \left (1-x^{2}+\frac {1}{4} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {3 x^5}{40}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {x^4}{4}-x^2+1\right ) \]

✓ Sympy. Time used: 0.221 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = - \frac {x^{5} r{\left (3 \right )}}{4} + C_{2} \left (\frac {x^{4}}{4} - x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{6}\right ) \]