64.15.1 problem 1

Internal problem ID [13507]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 1
Date solved : Monday, March 31, 2025 at 07:59:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-3 x \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: 0.029 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{5}/{3}} \left (1+\frac {17}{36} x +\frac {1241}{7128} x^{2}+\frac {80665}{1347192} x^{3}+\frac {972725}{48498912} x^{4}+\frac {5797441}{872980416} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{2} x -\frac {1}{2} x^{2}-\frac {5}{24} x^{3}-\frac {25}{336} x^{4}-\frac {17}{672} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 85

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

\[ y(x)\to c_2 \left (-\frac {17 x^5}{672}-\frac {25 x^4}{336}-\frac {5 x^3}{24}-\frac {x^2}{2}-\frac {x}{2}+1\right )+c_1 \left (\frac {5797441 x^5}{872980416}+\frac {972725 x^4}{48498912}+\frac {80665 x^3}{1347192}+\frac {1241 x^2}{7128}+\frac {17 x}{36}+1\right ) x^{5/3} \]

✓ Sympy. Time used: 0.890 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{2} x^{\frac {5}{3}} + C_{1} + O\left (x^{6}\right ) \]