20.26.30 problem 24

Internal problem ID [4055]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 05:23:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 46

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

\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (2 x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-2 x -2 x^{2}+\frac {2}{3} x^{3}-\frac {2}{9} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 48

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

\[ y(x)\to c_1 \left (\frac {2 \log (x)}{x}-\frac {2 x^4-6 x^3+18 x^2+36 x-9}{9 x^2}\right )+\frac {c_2}{x} \]

✓ Sympy. Time used: 0.836 (sec). Leaf size: 8

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

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