56.4.28 problem 24

Internal problem ID [8917]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 24
Date solved : Sunday, March 30, 2025 at 01:53:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 90

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

\[ y = \left (1+\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}+\frac {37 \left (x -1\right )^{5}}{480}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{24}+\frac {19 \left (x -1\right )^{4}}{192}-\frac {119 \left (x -1\right )^{5}}{1920}\right ) y^{\prime }\left (1\right )+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}+\frac {89 \left (x -1\right )^{5}}{960}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 105

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

\[ y(x)\to \frac {89}{960} (x-1)^5-\frac {3}{32} (x-1)^4+\frac {1}{12} (x-1)^3+c_1 \left (\frac {37}{480} (x-1)^5-\frac {5}{48} (x-1)^4+\frac {1}{6} (x-1)^3+1\right )+c_2 \left (-\frac {119 (x-1)^5}{1920}+\frac {19}{192} (x-1)^4-\frac {1}{24} (x-1)^3+\frac {1}{4} (x-1)^2+x-1\right ) \]

✗ Sympy

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

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