12.14.46 problem 48

Internal problem ID [1987]
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 : 48
Date solved : Tuesday, March 04, 2025 at 01:47:34 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (x^{2}+8\right ) y^{\prime \prime }+7 x \left (x^{2}+2\right ) y^{\prime }-\left (-9 x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*(x^2+8)*diff(diff(y(x),x),x)+7*x*(x^2+2)*diff(y(x),x)-(-9*x^2+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{4}} \left (1-\frac {13}{64} x^{2}+\frac {273}{8192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {1}{3} x^{2}+\frac {2}{33} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 50

ode=x^2*(8+x^2)*D[y[x],{x,2}]+7*x*(2+x^2)*D[y[x],x]-(2-9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {273 x^4}{8192}-\frac {13 x^2}{64}+1\right )+\frac {c_2 \left (\frac {2 x^4}{33}-\frac {x^2}{3}+1\right )}{x} \]

✓ Sympy. Time used: 1.125 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 8)*Derivative(y(x), (x, 2)) + 7*x*(x**2 + 2)*Derivative(y(x), x) - (2 - 9*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [4]{x} + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]