67.23.30 problem 33.11 (d)

Internal problem ID [16963]
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 : Thursday, October 02, 2025 at 01:40:43 PM
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*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 44

Order:=4; 
ode:=sin(x)*diff(diff(y(x),x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=1/2*Pi);
 

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

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

ode=Sin[x]*D[y[x],{x,2}]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Pi/2,3}]
 

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

✓ Sympy. Time used: 0.588 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + sin(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=pi/2,n=4)
 

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