73.26.3 problem 36.2 (c)

Internal problem ID [15743]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (c)
Date solved : Tuesday, January 28, 2025 at 08:06:25 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 32

Order:=6; 
dsolve(diff(y(x),x$2)+1/x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]