73.23.30 problem 33.11 (d)

Internal problem ID [15676]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.11 (d)
Date solved : Tuesday, January 28, 2025 at 08:04:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right ) y^{\prime \prime }-y&=0 \end{align*}

Using series method with expansion around

\begin{align*} \frac {\pi }{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 44

Order:=4; 
dsolve(sin(x)*diff(y(x),x$2)-y(x)=0,y(x),type='series',x=Pi/2);
 

\[ y = \left (1+\frac {\left (x -\frac {\pi }{2}\right )^{2}}{2}\right ) y \left (\frac {\pi }{2}\right )+\left (x -\frac {\pi }{2}+\frac {\left (x -\frac {\pi }{2}\right )^{3}}{6}\right ) y^{\prime }\left (\frac {\pi }{2}\right )+O\left (x^{4}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 45

AsymptoticDSolveValue[Sin[x]*D[y[x],{x,2}]-y[x]==0,y[x],{x,Pi/2,"4"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{2} \left (x-\frac {\pi }{2}\right )^2+1\right )+c_2 \left (\frac {1}{6} \left (x-\frac {\pi }{2}\right )^3+x-\frac {\pi }{2}\right ) \]