68.17.14 problem 14

Internal problem ID [17832]
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 : 14
Date solved : Thursday, October 02, 2025 at 02:28:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 46

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+2 x +2 x^{2}+\frac {11}{9} x^{3}+\frac {35}{72} x^{4}+\frac {103}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {64}{27} x^{3}-\frac {497}{432} x^{4}-\frac {9371}{27000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 110

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

\[ y(x)\to c_1 x \left (\frac {103 x^5}{900}+\frac {35 x^4}{72}+\frac {11 x^3}{9}+2 x^2+2 x+1\right )+c_2 \left (x \left (-\frac {9371 x^5}{27000}-\frac {497 x^4}{432}-\frac {64 x^3}{27}-3 x^2-2 x\right )+x \left (\frac {103 x^5}{900}+\frac {35 x^4}{72}+\frac {11 x^3}{9}+2 x^2+2 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.417 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {35 x^{4}}{72} + \frac {11 x^{3}}{9} + 2 x^{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]