87.26.8 problem 8

Internal problem ID [23842]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:45:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+3 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: 0.010 (sec). Leaf size: 40

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

\[ y = \frac {\left (9 x -\frac {9}{4} x^{2}-\frac {3}{4} x^{3}-\frac {9}{32} x^{4}-\frac {81}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-3 x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 66

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

\[ y(x)\to c_2 \left (\frac {-\frac {81 x^5}{800}-\frac {9 x^4}{32}-\frac {3 x^3}{4}-\frac {9 x^2}{4}+9 x}{x}+\frac {(1-3 x) \log (x)}{x}\right )+\frac {c_1 (1-3 x)}{x} \]

✓ Sympy. Time used: 0.287 (sec). Leaf size: 14

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

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