12.14.45 problem 47

Internal problem ID [1986]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 47
Date solved : Tuesday, March 04, 2025 at 01:47:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {6}{13} x^{2}+\frac {36}{325} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode=6*x^2*D[y[x],{x,2}]+x*(1+6*x^2)*D[y[x],x]+(1+9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {36 x^4}{325}-\frac {6 x^2}{13}+1\right )+c_2 \sqrt [3]{x} \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.002 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {36 x^{4}}{325} - \frac {6 x^{2}}{13} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{4}}{8} - \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]