12.14.32 problem 34

Internal problem ID [1973]
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 : 34
Date solved : Tuesday, March 04, 2025 at 01:47:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 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: 0.009 (sec). Leaf size: 35

Order:=6; 
ode:=8*x^2*(-x^2+1)*diff(diff(y(x),x),x)+2*x*(-13*x^2+1)*diff(y(x),x)+(-9*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1+\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1+\frac {5}{9} x^{2}+\frac {65}{153} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 52

ode=8*x^2*(1-x^2)*D[y[x],{x,2}]+2*x*(1-13*x^2)*D[y[x],x]+(1-9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {65 x^4}{153}+\frac {5 x^2}{9}+1\right )+c_2 \sqrt [4]{x} \left (\frac {3 x^4}{8}+\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.235 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(1 - x**2)*Derivative(y(x), (x, 2)) + 2*x*(1 - 13*x**2)*Derivative(y(x), x) + (1 - 9*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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