20.26.17 problem 11

Internal problem ID [4042]
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 : 11
Date solved : Tuesday, March 04, 2025 at 05:23:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 39

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {9}{10} x^{2}-\frac {4}{5} x^{3}+\frac {5}{7} x^{4}-\frac {9}{14} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 47

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

\[ y(x)\to c_2 \left (\frac {5 x^6}{7}-\frac {4 x^5}{5}+\frac {9 x^4}{10}-x^3+x^2\right )+c_1 \left (\frac {1}{x}+\frac {1}{2}\right ) \]

✓ Sympy. Time used: 0.987 (sec). Leaf size: 14

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

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