42.1.11 problem 3.24 (f)

Internal problem ID [6833]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.24 (f)
Date solved : Sunday, March 30, 2025 at 11:24:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }-\sin \left (x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 32

Order:=6; 
ode:=sin(x)*diff(diff(y(x),x),x)-2*cos(x)*diff(y(x),x)-sin(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1-\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (12-6 x^{2}+\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 44

ode=Sin[x]*D[y[x],{x,2}]-2*Cos[x]*D[y[x],x]-Sin[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^4}{24}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{280}-\frac {x^5}{10}+x^3\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*sin(x) + sin(x)*Derivative(y(x), (x, 2)) - 2*cos(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -y(x)*sin(x) + sin(x)*Derivative(y(x), (x, 2)) - 2*cos(x)*Derivative(y(x), x) does not match hint 2nd_power_series_regular