73.26.9 problem 36.2 (i)

Internal problem ID [15749]
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 : Tuesday, January 28, 2025 at 08:06:33 AM
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*}

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 42

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=0);
 

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 59

AsymptoticDSolveValue[x*D[y[x],{x,2}]+4*D[y[x],x]+12/(x+2)^2*y[x]==0,y[x],{x,0,"6"-1}]
 

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