67.26.9 problem 36.2 (i)

Internal problem ID [17036]
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 (i)
Date solved : Thursday, October 02, 2025 at 01:42:11 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*} 0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 42

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=0);
 

\[ y = c_1 \left (1-\frac {3}{4} x +\frac {21}{40} x^{2}-\frac {27}{80} x^{3}+\frac {33}{160} x^{4}-\frac {39}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12+18 x +9 x^{2}+\frac {9}{8} x^{4}-\frac {63}{80} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 59

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,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {3}{2 x^2}+\frac {3}{4 x}+\frac {1}{8}\right )+c_2 \left (\frac {33 x^4}{160}-\frac {27 x^3}{80}+\frac {21 x^2}{40}-\frac {3 x}{4}+1\right ) \]

✓ Sympy. Time used: 0.420 (sec). Leaf size: 39

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=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {54 x^{5}}{175} + \frac {36 x^{4}}{35} - \frac {12 x^{3}}{5} + \frac {18 x^{2}}{5} - 3 x + 1\right ) + O\left (x^{6}\right ) \]