20.26.20 problem 14

Internal problem ID [4045]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 14
Date solved : Monday, January 27, 2025 at 08:06:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 32

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

\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (1+\frac {3}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 33

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

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {x}{8}+\frac {3}{2 x}\right )+c_2 \left (\frac {1}{x^2}+\frac {1}{3}\right ) \]