52.2.3 problem 3

Internal problem ID [8254]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 3
Date solved : Monday, January 27, 2025 at 03:47:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 79

Order:=8; 
dsolve((x^2-9)^2*diff(y(x),x$2)+(x+3)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}+\frac {1732937}{1464449448420} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}-\frac {13738871}{488149816140} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 98

AsymptoticDSolveValue[(x^2-9)^2*D[y[x],{x,2}]+(x+3)*D[y[x],x]+2*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {1732937 x^7}{1464449448420}-\frac {194981 x^6}{7748409780}+\frac {304 x^5}{23914845}-\frac {289 x^4}{708588}+\frac {x^3}{6561}-\frac {x^2}{81}+1\right )+c_2 \left (-\frac {13738871 x^7}{488149816140}-\frac {78469 x^6}{2582803260}-\frac {596 x^5}{1594323}-\frac {131 x^4}{236196}-\frac {13 x^3}{2187}-\frac {x^2}{54}+x\right ) \]