64.14.16 problem 16

Internal problem ID [13504]
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 : 16
Date solved : Monday, March 31, 2025 at 07:59:50 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 76

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {\left (x -1\right )^{2}}{2}-\frac {5 \left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{6}-\frac {91 \left (x -1\right )^{5}}{60}\right ) y \left (1\right )+\left (x -1-\frac {3 \left (x -1\right )^{2}}{2}+\frac {13 \left (x -1\right )^{3}}{6}-\frac {35 \left (x -1\right )^{4}}{12}+\frac {56 \left (x -1\right )^{5}}{15}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 87

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (-\frac {91}{60} (x-1)^5+\frac {7}{6} (x-1)^4-\frac {5}{6} (x-1)^3+\frac {1}{2} (x-1)^2+1\right )+c_2 \left (\frac {56}{15} (x-1)^5-\frac {35}{12} (x-1)^4+\frac {13}{6} (x-1)^3-\frac {3}{2} (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.750 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {35 \left (x - 1\right )^{4}}{12} + \frac {13 \left (x - 1\right )^{3}}{6} - \frac {3 \left (x - 1\right )^{2}}{2} - 1\right ) + C_{1} \left (\frac {7 \left (x - 1\right )^{4}}{6} - \frac {5 \left (x - 1\right )^{3}}{6} + \frac {\left (x - 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]