73.24.19 problem 34.7 (e)

Internal problem ID [15699]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.7 (e)
Date solved : Tuesday, January 28, 2025 at 08:05:32 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}+\frac {\left (x -1\right )^{4}}{96}-\frac {\left (x -1\right )^{5}}{960}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}-\frac {\left (x -1\right )^{5}}{96}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (-\frac {1}{960} (x-1)^5+\frac {1}{96} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {1}{96} (x-1)^5+\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+x-1\right ) \]