Internal problem ID [14014]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page
739
Problem number: 36.2 (j).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (x +2\right )^{2}}=0} \] With the expansion point for the power series method at \(x = -2\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 47
Order:=6; dsolve(x*diff(y(x),x$2)+4*diff(y(x),x)+12/(x+2)^2*y(x)=0,y(x),type='series',x=-2);
\[ y \left (x \right ) = \frac {c_{1} \left (x +2\right )^{5} \left (1+\frac {3}{2} \left (x +2\right )+\frac {3}{2} \left (x +2\right )^{2}+\frac {5}{4} \left (x +2\right )^{3}+\frac {15}{16} \left (x +2\right )^{4}+\frac {21}{32} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )+c_{2} \left (2880+720 \left (x +2\right )+120 \left (x +2\right )^{2}+15 \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )}{\left (x +2\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 70
AsymptoticDSolveValue[x*y''[x]+4*y'[x]+12/(x+2)^2*y[x]==0,y[x],{x,-2,5}]
\[ y(x)\to c_1 \left (\frac {1}{4 (x+2)}+\frac {1}{(x+2)^2}+\frac {1}{24}\right )+c_2 \left (\frac {15}{16} (x+2)^7+\frac {5}{4} (x+2)^6+\frac {3}{2} (x+2)^5+\frac {3}{2} (x+2)^4+(x+2)^3\right ) \]