12.13.14 problem 14

Internal problem ID [1905]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 14
Date solved : Saturday, March 29, 2025 at 11:42:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3+x \right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-\left (2-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

Order:=6; 
ode:=(x+3)*diff(diff(y(x),x),x)+(2*x+1)*diff(y(x),x)-(2-x)*y(x) = 0; 
ic:=y(-1) = 1, D(y)(-1) = 0; 
dsolve([ode,ic],y(x),type='series',x=-1);
 

\[ y = 1+\frac {3}{4} \left (x +1\right )^{2}-\frac {1}{12} \left (x +1\right )^{3}-\frac {1}{48} \left (x +1\right )^{4}-\frac {1}{120} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 41

ode=(3+x)*D[y[x],{x,2}]+(1+2*x)*D[y[x],x]-(2-x)*y[x]==0; 
ic={y[-1]==1,Derivative[1][y][-1]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to -\frac {1}{120} (x+1)^5-\frac {1}{48} (x+1)^4-\frac {1}{12} (x+1)^3+\frac {3}{4} (x+1)^2+1 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 2)*y(x) + (x + 3)*Derivative(y(x), (x, 2)) + (2*x + 1)*Derivative(y(x), x),0) 
ics = {y(-1): 1, Subs(Derivative(y(x), x), x, -1): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=-1,n=6)
 

IndexError : list index out of range