12.13.11 problem 11

Internal problem ID [1902]
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 : 11
Date solved : Saturday, March 29, 2025 at 11:42:46 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

\[ y = -2+3 \left (1+x \right )-\frac {1}{2} \left (1+x \right )^{2}-\frac {2}{3} \left (1+x \right )^{3}+\frac {5}{8} \left (1+x \right )^{4}-\frac {11}{30} \left (1+x \right )^{5}+\operatorname {O}\left (\left (1+x \right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to -\frac {11}{30} (x+1)^5+\frac {5}{8} (x+1)^4-\frac {2}{3} (x+1)^3-\frac {1}{2} (x+1)^2+3 (x+1)-2 \]

✓ Sympy. Time used: 0.765 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {\left (x + 1\right )^{4}}{8} + \frac {\left (x + 1\right )^{3}}{3} - \frac {\left (x + 1\right )^{2}}{2} + 1\right ) + C_{1} \left (x + \frac {\left (x + 1\right )^{4}}{8} - \frac {\left (x + 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]