67.26.10 problem 36.2 (j)

Internal problem ID [17037]
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)
Date solved : Thursday, October 02, 2025 at 01:42:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (x +2\right )^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 48

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+4*diff(y(x),x)+12/(x+2)^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=-2);
 

\[ y = \frac {c_1 \left (x +2\right )^{5} \left (1+\frac {3}{2} \left (x +2\right )+\frac {3}{2} \left (x +2\right )^{2}+\frac {5}{4} \left (x +2\right )^{3}+\frac {15}{16} \left (x +2\right )^{4}+\frac {21}{32} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )+c_2 \left (2880+720 \left (x +2\right )+120 \left (x +2\right )^{2}+15 \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )}{\left (x +2\right )^{2}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 70

ode=x*D[y[x],{x,2}]+4*D[y[x],x]+12/(x+2)^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-2,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{4 (x+2)}+\frac {1}{(x+2)^2}+\frac {1}{24}\right )+c_2 \left (\frac {15}{16} (x+2)^7+\frac {5}{4} (x+2)^6+\frac {3}{2} (x+2)^5+\frac {3}{2} (x+2)^4+(x+2)^3\right ) \]

✓ Sympy. Time used: 0.418 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), x) + 12*y(x)/(x + 2)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=-2,n=6)
 

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