37.1.11 problem 7.2.102

Internal problem ID [8142]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.2.1 Exercises. page 290
Problem number : 7.2.102
Date solved : Tuesday, September 30, 2025 at 05:16:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-x y&=\frac {1}{1-x} \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: 16

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

\[ y = \frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {3}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 32

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

\[ y(x)\to \frac {3 x^5}{40}+\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + Derivative(y(x), (x, 2)) - 1/(1 - x),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*y(x) + Derivative(y(x), (x, 2)) - 1/(1 - x) does not match hint 2nd_power_series_regular