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 : Tuesday, March 04, 2025 at 01:54:34 PM
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*}

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

ode:=cos(y(t))*diff(y(t),t) = -t*sin(y(t))/(t^2+1); 
ic:=y(1) = 1/2*Pi; 
dsolve([ode,ic],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*}

✓ Mathematica. Time used: 15.08 (sec). Leaf size: 21

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

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

✓ Sympy. Time used: 0.457 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*sin(y(t))/(t**2 + 1) + cos(y(t))*Derivative(y(t), t),0) 
ics = {y(1): pi/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = \pi - \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {t^{2} + 1}} \right )}, \ y{\left (t \right )} = \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {t^{2} + 1}} \right )}\right ] \]