67.25.28 problem 35.4 (n)

Internal problem ID [17023]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (n)
Date solved : Thursday, October 02, 2025 at 01:42:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (9 x^{3}+9 x^{2}\right ) y^{\prime \prime }+\left (27 x^{2}+9 x \right ) y^{\prime }+\left (8 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 41

Order:=6; 
ode:=(9*x^3+9*x^2)*diff(diff(y(x),x),x)+(27*x^2+9*x)*diff(y(x),x)+(8*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 62

ode=(9*x^2+9*x^3)*D[y[x],{x,2}]+(9*x+27*x^2)*D[y[x],x]+(8*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.354 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((8*x - 1)*y(x) + (27*x**2 + 9*x)*Derivative(y(x), x) + (9*x**3 + 9*x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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