46.4.2 problem 2

Internal problem ID [7329]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.5. Bessel Functions Y(x). General Solution page 200
Problem number : 2
Date solved : Monday, January 27, 2025 at 02:50:29 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+5 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 44

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

\[ y = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{4}} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 47

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

\[ y(x)\to c_2 \left (\frac {x^4}{384}-\frac {x^2}{12}+1\right )+c_1 \left (\frac {\left (x^2+8\right )^2}{64 x^4}-\frac {\log (x)}{16}\right ) \]