6.16.26 problem 22

Internal problem ID [2088]
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 III. Exercises 7.7. Page 389
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 05:23:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }-x \left (9+8 x \right ) y^{\prime }-12 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)-x*(9+8*x)*diff(y(x),x)-12*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{10} \left (1-8 x +40 x^{2}-160 x^{3}+560 x^{4}-1792 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-1316818944000+1755758592000 x -2194698240000 x^{2}[phone] x^{3}-2508226560000 x^{4}[phone] x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 62

ode=x^2*(1+2*x)*D[y[x],{x,2}]-x*(9+8*x)*D[y[x],x]-12*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {40 x^4}{21}-\frac {40 x^3}{21}+\frac {5 x^2}{3}-\frac {4 x}{3}+1\right )+c_2 \left (560 x^{14}-160 x^{13}+40 x^{12}-8 x^{11}+x^{10}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) - x*(8*x + 9)*Derivative(y(x), x) - 12*x*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None