73.25.30 problem 35.5 (b)

Internal problem ID [15738]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.5 (b)
Date solved : Tuesday, January 28, 2025 at 08:06:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x +2}+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -2 \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 54

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

\[ y(x)\to c_1 \left (\frac {1}{24} (x+2)^3+\frac {1}{2} (-x-2)+\frac {1}{x+2}\right )+c_2 \left (\frac {1}{120} (x+2)^4-\frac {1}{6} (x+2)^2+1\right ) \]