54.6.2 problem 2

Internal problem ID [8634]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 2
Date solved : Wednesday, March 05, 2025 at 06:11:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+2 x \left (1+6 x \right ) y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 53

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

\[ y = c_{1} x \left (1-3 x +\frac {42}{5} x^{2}-\frac {112}{5} x^{3}+\frac {288}{5} x^{4}-144 x^{5}+352 x^{6}-\frac {4224}{5} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-72 x +288 x^{2}-960 x^{3}+2880 x^{4}-8064 x^{5}+21504 x^{6}-55296 x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 76

ode=x^2*(1+2*x)*D[y[x],{x,2}]+2*x*(1+6*x)*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (1792 x^4-672 x^3+240 x^2+\frac {1}{x^2}-80 x-\frac {6}{x}+24\right )+c_2 \left (352 x^7-144 x^6+\frac {288 x^5}{5}-\frac {112 x^4}{5}+\frac {42 x^3}{5}-3 x^2+x\right ) \]

✓ Sympy. Time used: 1.015 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{2} x + \frac {C_{1}}{x^{2}} + O\left (x^{8}\right ) \]