69.23.8 problem 731

Internal problem ID [18492]
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 : Friday, October 03, 2025 at 07:32:43 AM
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)+sin(y(x))*x = 0; 
ic:=[y(0) = 1/2*Pi, D(y)(0) = 0, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

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

✗ Mathematica

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}]
 

Not solved

✗ Sympy

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

Series solution not supported for ode of order > 2