12.16.20 problem 16

Internal problem ID [2082]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 16
Date solved : Tuesday, March 04, 2025 at 01:49:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (3+x \right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 46

Order:=6; 
ode:=3*x^2*(x+3)*diff(diff(y(x),x),x)-x*(15+x)*diff(y(x),x)-20*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {4}{9} x +\frac {13}{81} x^{2}-\frac {832}{15309} x^{3}+\frac {2470}{137781} x^{4}-\frac {21736}{3720087} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-\frac {64}{3} x +\frac {16}{27} x^{2}-\frac {112}{6561} x^{4}+\frac {448}{59049} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{3}}} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 85

ode=3*x^2*(3+x)*D[y[x],{x,2}]-x*(15+x)*D[y[x],x]-20*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {7 x^{10/3}}{59049}-\frac {x^{4/3}}{243}+\frac {1}{x^{2/3}}+\frac {4 \sqrt [3]{x}}{27}\right )+c_2 \left (\frac {2470 x^{22/3}}{137781}-\frac {832 x^{19/3}}{15309}+\frac {13 x^{16/3}}{81}-\frac {4 x^{13/3}}{9}+x^{10/3}\right ) \]

✓ Sympy. Time used: 1.004 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(x + 3)*Derivative(y(x), (x, 2)) - x*(x + 15)*Derivative(y(x), x) - 20*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {10}{3}} + \frac {C_{1}}{x^{\frac {2}{3}}} + O\left (x^{6}\right ) \]