20.26.35 problem 29

Internal problem ID [4060]
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 : 29
Date solved : Sunday, March 30, 2025 at 02:15:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 46

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

\[ y = \frac {\left (x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right ) x +\left (1-x -\frac {1}{2} x^{2}+\frac {1}{12} x^{3}-\frac {1}{72} x^{4}+\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{2}} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \left (\frac {\log (x)}{x}-\frac {x^4-6 x^3+36 x^2+144 x-72}{72 x^2}\right )+\frac {c_2}{x} \]

✓ Sympy. Time used: 0.792 (sec). Leaf size: 8

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

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