14.25.1 problem 1

Internal problem ID [4006]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 07:00:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{1-x}+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 49

Order:=6; 
ode:=diff(diff(y(x),x),x)+1/(1-x)*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_1 \left (\frac {x^5}{60}+\frac {x^4}{24}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {x^5}{24}-\frac {x^4}{12}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 61

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

\[ y{\left (x \right )} = \frac {x^{4} r{\left (3 \right )}}{4} + \frac {x^{5} r{\left (3 \right )}}{10} + C_{2} \left (\frac {11 x^{5}}{240} + \frac {x^{4}}{12} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{4}}{24} - \frac {x^{3}}{12} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]