14.26.31 problem 25

Internal problem ID [4056]
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 : 25
Date solved : Tuesday, September 30, 2025 at 07:01:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 36

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

\[ y = \left (\left (-x +\frac {1}{4} x^{2}-\frac {1}{18} x^{3}+\frac {1}{96} x^{4}-\frac {1}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

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

\[ y(x)\to c_2 \left (x \left (-\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^3}{18}+\frac {x^2}{4}-x\right )+x \log (x)\right )+c_1 x \]

✓ Sympy. Time used: 0.294 (sec). Leaf size: 8

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) + (1 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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