45.3.10 problem 10

Internal problem ID [7268]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 10
Date solved : Monday, January 27, 2025 at 02:49:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (2 x^{2}-64\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(2*x^2-64)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{8} \left (1-\frac {1}{18} x^{2}+\frac {1}{720} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-27360196043587190784000000-1954299717399085056000000 x^{2}-81429154891628544000000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{8}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 46

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(2*x^2-64)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^{12}}{720}-\frac {x^{10}}{18}+x^8\right )+c_1 \left (\frac {1}{x^8}+\frac {1}{14 x^6}+\frac {1}{336 x^4}\right ) \]