56.4.8 problem 8

Internal problem ID [8897]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:52:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y&=x^{4} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 43

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(-x^2+1)*y(x) = x^4; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{4} \left (\frac {1}{21}+\operatorname {O}\left (x^{2}\right )\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 150

ode=2*x^2*D[y[x],{x,2}] - x*D[y[x],x] + (1-x^2 )*y[x] ==x^4; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{55}-\frac {2 x^{7/2}}{7}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {x^5}{30}+\frac {x^3}{3}\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 + 2*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (1 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -x**4 + 2*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (1 - x**2)*y(x) does not match hint 2nd_power_series_regular