8.2.10 problem 10

Internal problem ID [2498]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 05:37:13 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*}

✓ Maple. Time used: 0.186 (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,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 13.265 (sec). Leaf size: 21

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

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

✓ Sympy. Time used: 0.313 (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)
 

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