6.15.3 problem Example 7.6.3 page 370

Internal problem ID [2001]
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 : Example 7.6.3 page 370
Date solved : Tuesday, September 30, 2025 at 05:22:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 34

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {3}{4} x^{2}+\frac {15}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{8} x^{2}-\frac {13}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 65

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

\[ y(x)\to c_1 x \left (\frac {15 x^4}{32}+\frac {3 x^2}{4}+1\right )+c_2 \left (x \left (-\frac {13 x^4}{128}-\frac {x^2}{8}\right )+x \left (\frac {15 x^4}{32}+\frac {3 x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.510 (sec). Leaf size: 8

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

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