30.13.14 problem 15

Internal problem ID [7646]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 04:55:26 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (-\frac {x^4}{6}-\frac {x^3}{6}+\frac {x^2}{2}+x\right )+c_1 \left (\frac {x^5}{20}+\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.245 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*Derivative(y(x), x) + 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)
 

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