75.23.8 problem 731

Internal problem ID [17205]
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 : Tuesday, January 28, 2025 at 08:27:32 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*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
dsolve([diff(y(x),x$3)+x*sin(y(x))=0,y(0) = 1/2*Pi, D(y)(0) = 0, (D@@2)(y)(0) = 0],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],{x,3}]+x*Sin[y[x]]==0,{y[0]==Pi/2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}},y[x],{x,0,"6"-1}]
 

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