40.2.23 problem 48

Internal problem ID [6601]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary problems. Page 22
Problem number : 48
Date solved : Sunday, March 30, 2025 at 11:11:46 AM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right )+\left (1+{\mathrm e}^{-x}\right ) \sin \left (y\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{4} \end{align*}

✓ Maple. Time used: 0.336 (sec). Leaf size: 14

ode:=cos(y(x))+(1+exp(-x))*sin(y(x))*diff(y(x),x) = 0; 
ic:=y(0) = 1/4*Pi; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \arccos \left (\frac {\sqrt {2}\, \left ({\mathrm e}^{x}+1\right )}{4}\right ) \]

✓ Mathematica. Time used: 52.488 (sec). Leaf size: 20

ode=Cos[y[x]]+(1+Exp[-x])*Sin[y[x]]*D[y[x],x]== 0; 
ic={y[0]==Pi/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \arccos \left (\frac {e^x+1}{2 \sqrt {2}}\right ) \]

✓ Sympy. Time used: 0.611 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 + exp(-x))*sin(y(x))*Derivative(y(x), x) + cos(y(x)),0) 
ics = {y(0): pi/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \operatorname {acos}{\left (\frac {\sqrt {2} e^{x}}{4} + \frac {\sqrt {2}}{4} \right )} \]