Internal problem ID [13620]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sin \left (x \right ) y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = \frac {\pi }{2}\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
Order:=4; dsolve(sin(x)*diff(y(x),x$2)-y(x)=0,y(x),type='series',x=Pi/2);
\[ y \left (x \right ) = \left (1+\frac {\left (x -\frac {\pi }{2}\right )^{2}}{2}\right ) y \left (\frac {\pi }{2}\right )+\left (x -\frac {\pi }{2}+\frac {\left (x -\frac {\pi }{2}\right )^{3}}{6}\right ) D\left (y \right )\left (\frac {\pi }{2}\right )+O\left (x^{4}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 45
AsymptoticDSolveValue[Sin[x]*y''[x]-y[x]==0,y[x],{x,Pi/2,3}]
\[ y(x)\to c_1 \left (\frac {1}{2} \left (x-\frac {\pi }{2}\right )^2+1\right )+c_2 \left (\frac {1}{6} \left (x-\frac {\pi }{2}\right )^3+x-\frac {\pi }{2}\right ) \]