69.23.11 problem 734

Internal problem ID [18495]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 734
Date solved : Thursday, October 02, 2025 at 03:14:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x y^{\prime }+y&=1 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 12

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 1; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = \frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 18

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

\[ y(x)\to \frac {x^4}{24}+\frac {x^2}{2} \]

✗ Sympy

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

ValueError : ODE -x*Derivative(y(x), x) + y(x) + Derivative(y(x), (x, 2)) - 1 does not match hint 2nd_power_series_regular