15.22.15 problem 15

Internal problem ID [3349]
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 : 15
Date solved : Monday, January 27, 2025 at 07:34:39 AM
CAS classification : [NONE]

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

Using series method with expansion around

\begin{align*} \frac {\pi }{2} \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1\\ y^{\prime }\left (\frac {\pi }{2}\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 22

Order:=5; 
dsolve([diff(y(x),x$2)=sin(x*y(x)),y(1/2*Pi) = 1, D(y)(1/2*Pi) = 1],y(x),type='series',x=Pi/2);
 

\[ y \left (x \right ) = 1+\left (-\frac {\pi }{2}+x \right )+\frac {1}{2} \left (-\frac {\pi }{2}+x \right )^{2}-\frac {1}{96} \left (\pi +2\right )^{2} \left (-\frac {\pi }{2}+x \right )^{4}+\operatorname {O}\left (\left (-\frac {\pi }{2}+x \right )^{5}\right ) \]

✓ Solution by Mathematica

Time used: 0.172 (sec). Leaf size: 47

AsymptoticDSolveValue[{D[y[x],{x,2}]==Sin[x*y[x]],{y[Pi/2]==1,Derivative[1][y][Pi/2]==1}},y[x],{x,Pi/2,"5"-1}]
 

\[ y(x)\to \frac {1}{96} \left (-4-4 \pi -\pi ^2\right ) \left (x-\frac {\pi }{2}\right )^4+\frac {1}{2} \left (x-\frac {\pi }{2}\right )^2+x-\frac {\pi }{2}+1 \]