15.25.12 problem 11

Internal problem ID [3399]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:39:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=6 \left (-x^{2}+1\right )^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

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

\[ y = \left (x^{2}+1\right ) y \left (0\right )-x^{4}+x y^{\prime }\left (0\right )+3 x^{2}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 26

ode=(1-x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]-2*y[x]==6*(1-x^2)^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -x^4+3 x^2+c_1 \left (x^2+1\right )+c_2 x \]

✗ Sympy

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

ValueError : ODE 2*x*Derivative(y(x), x) - 6*(1 - x**2)**2 + (1 - x**2)*Derivative(y(x), (x, 2)) - 2*y(x) does not match hint 2nd_power_series_regular