Internal problem ID [2378]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
\[ \boxed {y^{\prime \prime }-\sin \left (y x \right )=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = 1\right ] \end {align*}
With the expansion point for the power series method at \(x = \frac {\pi }{2}\).
✓ Solution by Maple
Time used: 0.0 (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: 0.165 (sec). Leaf size: 47
AsymptoticDSolveValue[{y''[x]==Sin[x*y[x]],{y[Pi/2]==1,y'[Pi/2]==1}},y[x],{x,Pi/2,4}]
\[ y(x)\to \frac {1}{96} \left (-4-4 \pi -\pi ^2\right ) \left (x-\frac {\pi }{2}\right )^4+\frac {1}{2} \left (x-\frac {\pi }{2}\right )^2+x-\frac {\pi }{2}+1 \]