54.3.8 problem 8

Internal problem ID [8564]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 8
Date solved : Monday, January 27, 2025 at 04:16:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

Order:=8; 
dsolve((x^2+4)*diff(y(x),x$2)+2*x*diff(y(x),x)-12*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {3}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {5}{12} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 42

AsymptoticDSolveValue[(x^2+4)*D[y[x],{x,2}]+2*x*D[y[x],x]-12*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {5 x^3}{12}+x\right )+c_1 \left (-\frac {x^6}{80}+\frac {3 x^4}{16}+\frac {3 x^2}{2}+1\right ) \]