7.15.14 problem 14

Internal problem ID [470]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 14
Date solved : Monday, January 27, 2025 at 02:53:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x^{2}+9\right ) y^{\prime }+\left (x^{2}+4\right ) 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: 59

Order:=6; 
dsolve((x^2-9)^2*diff(y(x),x$2)+(x^2+9)*diff(y(x),x)+(x^2+4)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {2}{81} x^{2}+\frac {2}{2187} x^{3}-\frac {49}{26244} x^{4}+\frac {463}{3542940} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{18} x^{2}-\frac {1}{162} x^{3}-\frac {47}{17496} x^{4}-\frac {697}{787320} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 70

AsymptoticDSolveValue[(x^2-9)^2*D[y[x],{x,2}]+(x^2+9)*D[y[x],x]+(x^2+4)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {463 x^5}{3542940}-\frac {49 x^4}{26244}+\frac {2 x^3}{2187}-\frac {2 x^2}{81}+1\right )+c_2 \left (-\frac {697 x^5}{787320}-\frac {47 x^4}{17496}-\frac {x^3}{162}-\frac {x^2}{18}+x\right ) \]