15.22.9 problem 9

Internal problem ID [3343]
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 : 9
Date solved : Monday, January 27, 2025 at 07:34:34 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\cos \left (x \right )+\sin \left (y\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 )&=\frac {\pi }{2} \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 18

Order:=4; 
dsolve([diff(y(x),x)=cos(x)+sin(y(x)),y(1/2*Pi) = 1/2*Pi],y(x),type='series',x=Pi/2);
 

\[ y \left (x \right ) = \frac {\pi }{2}+\left (-\frac {\pi }{2}+x \right )-\frac {1}{2} \left (-\frac {\pi }{2}+x \right )^{2}-\frac {1}{6} \left (-\frac {\pi }{2}+x \right )^{3}+\operatorname {O}\left (\left (-\frac {\pi }{2}+x \right )^{4}\right ) \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 22

AsymptoticDSolveValue[{D[y[x],x]==Cos[x]*Sin[y[x]],{y[Pi/2]==Pi/2}},y[x],{x,Pi/2,"4"-1}]
 

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