14.25.20 problem 21

Internal problem ID [4025]
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 11.4. page 758
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 07:00:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 38

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(3-2*x)*diff(y(x),x)+(1-2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (2 x +x^{2}+\frac {4}{9} x^{3}+\frac {1}{6} x^{4}+\frac {4}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*(3-2*x)*D[y[x],x]+(1-2*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {\frac {4 x^5}{75}+\frac {x^4}{6}+\frac {4 x^3}{9}+x^2+2 x}{x}+\frac {\log (x)}{x}\right )+\frac {c_1}{x} \]

✓ Sympy. Time used: 0.308 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(3 - 2*x)*Derivative(y(x), x) + (1 - 2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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