3.10 problem 10

Internal problem ID [1677]

Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 1.4. Page 24
Problem number: 10.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_separable]

Solve \begin {gather*} \boxed {\cos \relax (y) y^{\prime }+\frac {t \sin \relax (y)}{t^{2}+1}=0} \end {gather*} With initial conditions \begin {align*} \left [y \relax (1) = \frac {\pi }{2}\right ] \end {align*}

Solution by Maple

Time used: 0.157 (sec). Leaf size: 32

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

\[ y \relax (t ) = \arcsin \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right )+2 \arccos \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right ) \_B3 \]

Solution by Mathematica

Time used: 22.063 (sec). Leaf size: 21

DSolve[{Cos[y[t]]*y'[t] == -t*Sin[y[t]]/(t^2+1),y[1]==Pi/2},y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \text {ArcSin}\left (\frac {\sqrt {2}}{\sqrt {t^2+1}}\right ) \\ \end{align*}