12.15.39 problem 35

Internal problem ID [2037]
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 II. Exercises 7.6. Page 374
Problem number : 35
Date solved : Tuesday, March 04, 2025 at 01:48:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

Order:=6; 
ode:=3*x^2*(x^2+3)*diff(diff(y(x),x),x)+x*(11*x^2+3)*diff(y(x),x)+(5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{{1}/{3}} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {2}{9} x^{2}+\frac {5}{81} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{18} x^{2}+\frac {1}{54} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 42

ode=3*x^2*(3+x^2)*D[y[x],x]+x*(3+11*x^2)*D[y[x],x]+(1+5*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {1139 x^5}{405}+\frac {53 x^4}{81}+\frac {7 x^3}{9}-\frac {11 x^2}{9}+x+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 1.091 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(x**2 + 3)*Derivative(y(x), (x, 2)) + x*(11*x**2 + 3)*Derivative(y(x), x) + (5*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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