6.15.64 problem 65

Internal problem ID [2062]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 65
Date solved : Tuesday, September 30, 2025 at 05:23:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 32

Order:=6; 
ode:=9*x^2*(x^2+x+1)*diff(diff(y(x),x),x)+3*x*(13*x^2+7*x+1)*diff(y(x),x)+(25*x^2+4*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 48

ode=9*x^2*(1+x+x^2)*D[y[x],{x,2}]+3*x*(1+7*x+13*x^2)*D[y[x],x]+(1+4*x+25*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.464 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x**2 + x + 1)*Derivative(y(x), (x, 2)) + 3*x*(13*x**2 + 7*x + 1)*Derivative(y(x), x) + (25*x**2 + 4*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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