78.3.15 problem 4.b

Internal problem ID [21011]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 4.b
Date solved : Thursday, October 02, 2025 at 07:01:23 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 107

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

\[ y(x)\to c_1 \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (-\frac {137 x^5}{7200}-\frac {25 x^4}{288}-\frac {11 x^3}{36}-\frac {3 x^2}{4}+\left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \log (x)-x\right ) \]

✓ Sympy. Time used: 0.265 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{120} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + O\left (x^{6}\right ) \]