28.5.19 problem 9.19

Internal problem ID [4606]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.19
Date solved : Tuesday, March 04, 2025 at 06:56:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 47

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (\left (-6\right ) x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+18 x +18 x^{2}+11 x^{3}-\frac {3}{2} x^{4}-\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 50

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

\[ y(x)\to c_2 x^2+c_1 \left (-\frac {1}{2} x^2 \log (x)-\frac {3 x^4-26 x^3-36 x^2-36 x-24}{24 x}\right ) \]

✓ Sympy. Time used: 0.814 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*Derivative(y(x), x) + x**2*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)
 

\[ y{\left (x \right )} = C_{1} x^{2} + O\left (x^{6}\right ) \]