78.3.12 problem 2.g

Internal problem ID [21008]
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 : 2.g
Date solved : Thursday, October 02, 2025 at 07:01:22 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 62

Order:=5; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 69

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,4}]
 

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

✓ Sympy. Time used: 0.212 (sec). Leaf size: 54

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

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