20.26.8 problem (e)

Internal problem ID [4033]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : (e)
Date solved : Tuesday, March 04, 2025 at 05:23:10 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 }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 58

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

\[ y \left (x \right ) = c_{1} x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}+\frac {13}{288} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 80

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

\[ y(x)\to c_1 \left (\frac {1}{6} x \left (x^3-3 x^2+6 x-6\right ) \log (x)+\frac {1}{36} \left (-11 x^4+27 x^3-36 x^2+36\right )\right )+c_2 \left (\frac {x^5}{24}-\frac {x^4}{6}+\frac {x^3}{2}-x^2+x\right ) \]

✓ Sympy. Time used: 0.857 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{6}\right ) \]