67.26.6 problem 36.2 (f)

Internal problem ID [17033]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (f)
Date solved : Thursday, October 02, 2025 at 01:42:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {4}{5} x +\frac {4}{5} x^{2}-\frac {116}{105} x^{3}+\frac {311}{210} x^{4}-\frac {358}{175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (576 x^{4}-\frac {2304}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-576 x -576 x^{2}-192 x^{3}-384 x^{4}+\frac {15744}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 77

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

\[ y(x)\to c_1 \left (\frac {19 x^4+4 x^3+12 x^2+12 x+3}{3 x^2}-4 x^2 \log (x)\right )+c_2 \left (\frac {311 x^6}{210}-\frac {116 x^5}{105}+\frac {4 x^4}{5}-\frac {4 x^3}{5}+x^2\right ) \]

✓ Sympy. Time used: 0.422 (sec). Leaf size: 15

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

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