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 : Monday, January 27, 2025 at 05:38:06 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{(2+x)*D[y[x],{x,2}]+(2+x)*D[y[x],x]+y[x]==0,{y[-1]==-2,Derivative[1][y][-1]==3}},y[x],{x,-1,"6"-1}]
 

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