56.4.5 problem 5

Internal problem ID [8894]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 5
Date solved : Sunday, March 30, 2025 at 01:52:38 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^{2}+x +1 \end{align*}

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 224

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+\sqrt {x} \left (-\frac {79 x^{11/2}}{154440}-\frac {x^{9/2}}{1620}-\frac {37 x^{7/2}}{1260}-\frac {x^{5/2}}{25}-\frac {11 x^{3/2}}{15}-2 \sqrt {x}+\frac {2}{\sqrt {x}}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \left (\frac {x^6}{66528}+\frac {67 x^5}{55440}+\frac {x^4}{672}+\frac {29 x^3}{504}+\frac {x^2}{12}+\frac {7 x}{6}-\frac {1}{x}+\log (x)\right ) \]

✗ Sympy

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

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