12.15.60 problem 61

Internal problem ID [2058]
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 : 61
Date solved : Tuesday, March 04, 2025 at 01:49:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 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: 29

Order:=6; 
ode:=16*x^2*(x^2+1)*diff(diff(y(x),x),x)+8*x*(9*x^2+1)*diff(y(x),x)+(49*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{{1}/{4}} \left (x^{4}-x^{2}+1\right ) \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 42

ode=16*x^2*(1+x^2)*D[y[x],{x,2}]+8*x*(1+9*x^2)*D[y[x],x]+(1+49*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (x^4-x^2+1\right )+c_2 \sqrt [4]{x} \left (x^4-x^2+1\right ) \log (x) \]

✓ Sympy. Time used: 0.952 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 8*x*(9*x**2 + 1)*Derivative(y(x), x) + (49*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt [4]{x} + O\left (x^{6}\right ) \]