12.15.62 problem 63

Internal problem ID [2060]
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 : 63
Date solved : Tuesday, March 04, 2025 at 01:49:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (3+2 x \right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 44

Order:=6; 
ode:=4*x^2*(x^2+3*x+1)*diff(diff(y(x),x),x)+8*x^2*(2*x+3)*diff(y(x),x)+(9*x^2+3*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (-144 x^{5}+55 x^{4}-21 x^{3}+8 x^{2}-3 x +1\right ) \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 72

ode=4*x^2*(1+3*x+x^2)*D[y[x],{x,2}]+8*x^2*(3+2*x)*D[y[x],x]+(1+3*x+9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-144 x^5+55 x^4-21 x^3+8 x^2-3 x+1\right )+c_2 \sqrt {x} \left (-144 x^5+55 x^4-21 x^3+8 x^2-3 x+1\right ) \log (x) \]

✓ Sympy. Time used: 1.028 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(2*x + 3)*Derivative(y(x), x) + 4*x**2*(x**2 + 3*x + 1)*Derivative(y(x), (x, 2)) + (9*x**2 + 3*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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