6.16.15 problem 11

Internal problem ID [2077]
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 : 11
Date solved : Tuesday, September 30, 2025 at 05:23:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (3+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 37

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

\[ y = c_1 \,x^{3} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400+103680 x -64800 x^{2}+28800 x^{3}-10800 x^{4}+4320 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 39

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

\[ y(x)\to c_2 x^3+c_1 \left (\frac {1}{x^3}-\frac {6}{5 x^2}+\frac {x}{8}+\frac {3}{4 x}-\frac {1}{3}\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - (3*x + 9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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