Internal problem ID [13682]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{x +2}+y=0} \] With the expansion point for the power series method at \(x = -2\).
✓ Solution by Maple
Time used: 0.0 (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);
\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} \left (x +2\right )^{2}+\frac {1}{120} \left (x +2\right )^{4}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} \left (x +2\right )^{2}+\frac {1}{24} \left (x +2\right )^{4}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )}{x +2} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 54
AsymptoticDSolveValue[y''[x]+2/(x+2)*y'[x]+y[x]==0,y[x],{x,-2,5}]
\[ y(x)\to c_1 \left (\frac {1}{24} (x+2)^3+\frac {1}{2} (-x-2)+\frac {1}{x+2}\right )+c_2 \left (\frac {1}{120} (x+2)^4-\frac {1}{6} (x+2)^2+1\right ) \]