58.2.44 problem 44

Internal problem ID [9167]
Book : Second order enumerated odes
Section : section 2
Problem number : 44
Date solved : Monday, January 27, 2025 at 05:51:35 PM
CAS classification : [_Bessel]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-5\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 89

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-5)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{-\sqrt {5}} \left (1+\frac {1}{-4+4 \sqrt {5}} x^{2}+\frac {1}{32} \frac {1}{\left (-2+\sqrt {5}\right ) \left (\sqrt {5}-1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {5}} \left (1-\frac {1}{4+4 \sqrt {5}} x^{2}+\frac {1}{32} \frac {1}{\left (\sqrt {5}+2\right ) \left (\sqrt {5}+1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 210

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-5)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{\left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-1-\sqrt {5}+\left (3-\sqrt {5}\right ) \left (4-\sqrt {5}\right )\right )}-\frac {x^2}{-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )}+1\right ) x^{-\sqrt {5}}+c_1 \left (\frac {x^4}{\left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-1+\sqrt {5}+\left (3+\sqrt {5}\right ) \left (4+\sqrt {5}\right )\right )}-\frac {x^2}{-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )}+1\right ) x^{\sqrt {5}} \]