12.14.31 problem 33

Internal problem ID [1972]
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 : 33
Date solved : Tuesday, March 04, 2025 at 01:47:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{112} x^{2}+\frac {3}{17920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {1}{72} x^{2}+\frac {5}{19584} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 52

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {5 x^4}{19584}-\frac {x^2}{72}+1\right )+c_2 \sqrt [4]{x} \left (\frac {3 x^4}{17920}-\frac {x^2}{112}+1\right ) \]

✓ Sympy. Time used: 0.965 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {5 x^{4}}{19584} - \frac {x^{2}}{72} + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {3 x^{4}}{17920} - \frac {x^{2}}{112} + 1\right ) + O\left (x^{6}\right ) \]