13.3.10 problem 10

Internal problem ID [2327]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.4. Page 24
Problem number : 10
Date solved : Monday, January 27, 2025 at 05:45:55 AM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right ) y^{\prime }&=-\frac {t \sin \left (y\right )}{t^{2}+1} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.127 (sec). Leaf size: 35

dsolve([cos(y(t))*diff(y(t),t) = -t*sin(y(t))/(t^2+1),y(1) = 1/2*Pi],y(t), singsol=all)
 

\begin{align*} y &= \arcsin \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right ) \\ y &= -\arcsin \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right )+\pi \\ \end{align*}

✓ Solution by Mathematica

Time used: 15.080 (sec). Leaf size: 21

DSolve[{Cos[y[t]]*D[y[t],t] == -t*Sin[y[t]]/(t^2+1),y[1]==Pi/2},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \arcsin \left (\frac {\sqrt {2}}{\sqrt {t^2+1}}\right ) \]