14.26.2 problem Example 11.5.4 page 765

Internal problem ID [4027]
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 11.5. page 771
Problem number : Example 11.5.4 page 765
Date solved : Tuesday, September 30, 2025 at 07:00:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (3-x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 40

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(3-x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (3 x -\frac {1}{4} x^{2}-\frac {1}{36} x^{3}-\frac {1}{288} x^{4}-\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 66

ode=x^2*D[y[x],{x,2}]+x*(3-x)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {-\frac {x^5}{2400}-\frac {x^4}{288}-\frac {x^3}{36}-\frac {x^2}{4}+3 x}{x}+\frac {(1-x) \log (x)}{x}\right )+\frac {c_1 (1-x)}{x} \]

✓ Sympy. Time used: 0.264 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(3 - 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)
 

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