75.23.8 problem 731

Internal problem ID [17134]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 731
Date solved : Monday, March 31, 2025 at 03:42:56 PM
CAS classification : [NONE]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{2}\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

Order:=6; 
ode:=diff(diff(diff(y(x),x),x),x)+x*sin(y(x)) = 0; 
ic:=y(0) = 1/2*Pi, D(y)(0) = 0, (D@@2)(y)(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = \frac {\pi }{2}-\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 16

ode=D[y[x],{x,3}]+x*Sin[y[x]]==0; 
ic={y[0]==Pi/2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {\pi }{2}-\frac {x^4}{24} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - x + y(x)*sin(x),0) 
ics = {y(pi): 1, Subs(Derivative(y(x), x), x, pi): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=pi,n=6)
 

Series solution not supported for ode of order > 2