46.4.5 problem 5

Internal problem ID [7332]
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 : 5
Date solved : Monday, January 27, 2025 at 02:50:33 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+4 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 44

Order:=6; 
dsolve(4*x*diff(y(x),x$2)+4*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x +\frac {1}{64} x^{2}-\frac {1}{2304} x^{3}+\frac {1}{147456} x^{4}-\frac {1}{14745600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x -\frac {3}{64} x^{2}+\frac {11}{6912} x^{3}-\frac {25}{884736} x^{4}+\frac {137}{442368000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 117

AsymptoticDSolveValue[4*x*D[y[x],{x,2}]+4*D[y[x],x]+y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right )+c_2 \left (\frac {137 x^5}{442368000}-\frac {25 x^4}{884736}+\frac {11 x^3}{6912}-\frac {3 x^2}{64}+\left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right ) \log (x)+\frac {x}{2}\right ) \]