15.22.13 problem 13

Internal problem ID [3347]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 13
Date solved : Monday, January 27, 2025 at 07:34:37 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{4}\\ y^{\prime }\left (0\right )&=0 \end{align*}

Solution by Maple

Time used: 0.001 (sec). Leaf size: 26

Order:=7; 
dsolve([diff(y(x),x$2)=sin(y(x)),y(0) = 1/4*Pi, D(y)(0) = 0],y(x),type='series',x=0);
 
\[ y \left (x \right ) = \frac {\pi }{4}+\frac {1}{4} \sqrt {2}\, x^{2}+\frac {1}{48} x^{4}-\frac {1}{1440} \sqrt {2}\, x^{6}+\operatorname {O}\left (x^{7}\right ) \]

Solution by Mathematica

Time used: 0.156 (sec). Leaf size: 40

AsymptoticDSolveValue[{D[y[x],{x,2}]==Sin[y[x]],{y[0]==Pi/4,Derivative[1][y][0] ==0}},y[x],{x,0,"7"-1}]
 
\[ y(x)\to -\frac {x^6}{720 \sqrt {2}}+\frac {x^4}{48}+\frac {x^2}{2 \sqrt {2}}+\frac {\pi }{4} \]