54.3.21 problem 21

Internal problem ID [8577]
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 : 21
Date solved : Monday, January 27, 2025 at 04:16:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }-9 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)+x*diff(y(x),x)-9*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {9}{8} x^{2}+\frac {15}{128} x^{4}-\frac {7}{1024} x^{6}\right ) y \left (0\right )+\left (\frac {1}{3} x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[(x^2+4)*D[y[x],{x,2}]+x*D[y[x],x]-9*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^3}{3}+x\right )+c_1 \left (-\frac {7 x^6}{1024}+\frac {15 x^4}{128}+\frac {9 x^2}{8}+1\right ) \]