78.3.16 problem 4.c

Internal problem ID [21012]
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.c
Date solved : Thursday, October 02, 2025 at 07:01:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 46

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*(1-x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 114

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

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

✓ Sympy. Time used: 0.266 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(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)
 

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