56.4.53 problem 50

Internal problem ID [8942]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 50
Date solved : Monday, January 27, 2025 at 05:20:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}-8\right ) y&=0 \end{align*}

Using series method with expansion around

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

Solution by Maple

Time used: 0.013 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*diff(y(x), x$2)-x*diff(y(x), x)+(x^2-8)*y(x) = 0,y(x),type='series',x=0);
 
\[ y = c_{1} x^{4} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 42

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*D[y[x],x]+(x^2-8)*y[x]==0,y[x],{x,0,"6"-1}]
 
\[ y(x)\to c_1 \left (\frac {x^2}{64}+\frac {1}{x^2}+\frac {1}{8}\right )+c_2 \left (\frac {x^8}{640}-\frac {x^6}{16}+x^4\right ) \]