74.17.2 problem 2

Internal problem ID [16462]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 2
Date solved : Monday, March 31, 2025 at 02:53:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x +1\right ) y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}+5 y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x*(1+x)*diff(diff(y(x),x),x)+diff(y(x),x)/x^2+5*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 73

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

\[ y(x)\to c_1 \left (2 x^5-\frac {5 x^3}{3}+1\right )+c_2 e^{\frac {1}{2 x^2}-\frac {1}{x}} \left (\frac {272 x^5}{3}+\frac {88 x^4}{3}+\frac {35 x^3}{3}+4 x^2+2 x+1\right ) x^2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) + 5*y(x) + Derivative(y(x), x)/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x*(x + 1)*Derivative(y(x), (x, 2)) + 5*y(x) + Derivative(y(x), x)/x**2 does not match hint 2nd_power_series_regular