20.25.13 problem 14

Internal problem ID [4018]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 14
Date solved : Tuesday, March 04, 2025 at 05:22:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 47

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)-x*(x+8)*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{2}/{3}} \left (1-\frac {1}{6} x +\frac {5}{36} x^{2}+\frac {5}{81} x^{3}+\frac {11}{972} x^{4}+\frac {77}{58320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{3} \left (1+\frac {3}{10} x +\frac {3}{65} x^{2}+\frac {1}{208} x^{3}+\frac {3}{7904} x^{4}+\frac {21}{869440} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 88

ode=3*x^2*D[y[x],{x,2}]-x*(x+8)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {21 x^5}{869440}+\frac {3 x^4}{7904}+\frac {x^3}{208}+\frac {3 x^2}{65}+\frac {3 x}{10}+1\right ) x^3+c_2 \left (\frac {77 x^5}{58320}+\frac {11 x^4}{972}+\frac {5 x^3}{81}+\frac {5 x^2}{36}-\frac {x}{6}+1\right ) x^{2/3} \]

✓ Sympy. Time used: 0.909 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) - x*(x + 8)*Derivative(y(x), x) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{3} \left (\frac {3 x^{2}}{65} + \frac {3 x}{10} + 1\right ) + C_{1} x^{\frac {2}{3}} \left (\frac {11 x^{4}}{972} + \frac {5 x^{3}}{81} + \frac {5 x^{2}}{36} - \frac {x}{6} + 1\right ) + O\left (x^{6}\right ) \]