Internal problem ID [4911]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {4 x y^{\prime \prime }+4 y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 59
Order:=6; dsolve(4*x*diff(y(x),x$2)+4*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} x +\frac {1}{64} x^{2}-\frac {1}{2304} x^{3}+\frac {1}{147456} x^{4}-\frac {1}{14745600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x -\frac {3}{64} x^{2}+\frac {11}{6912} x^{3}-\frac {25}{884736} x^{4}+\frac {137}{442368000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 117
AsymptoticDSolveValue[4*x*y''[x]+4*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right )+c_2 \left (\frac {137 x^5}{442368000}-\frac {25 x^4}{884736}+\frac {11 x^3}{6912}-\frac {3 x^2}{64}+\left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right ) \log (x)+\frac {x}{2}\right ) \]