58.14.10 problem 10

Internal problem ID [14850]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:55:46 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x +3\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

Order:=6; 
ode:=(x+3)*diff(diff(y(x),x),x)+(x+2)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{2}+\frac {1}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{3240} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{2}+\frac {1}{36} x^{4}-\frac {1}{108} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

ode=(x+3)*D[y[x],{x,2}]+(x+2)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{108}+\frac {x^4}{36}-\frac {x^2}{3}+x\right )+c_1 \left (-\frac {7 x^5}{3240}-\frac {x^4}{216}+\frac {x^3}{18}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.271 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + 2)*Derivative(y(x), x) + (x + 3)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4}}{216} + \frac {x^{3}}{18} - \frac {x^{2}}{6} + 1\right ) + C_{1} x \left (\frac {x^{3}}{36} - \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]