32.6.5 problem Exercise 12.5, page 103

Internal problem ID [5870]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.5, page 103
Date solved : Sunday, March 30, 2025 at 10:20:47 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 14

ode:=diff(y(x),x)*sin(y(x))+sin(x)*cos(y(x)) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arccos \left ({\mathrm e}^{-\cos \left (x \right )} c_1 +1\right ) \]

✓ Mathematica. Time used: 0.734 (sec). Leaf size: 81

ode=D[y[x],x]*Sin[y[x]]+Sin[x]*Cos[y[x]]==Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to 0 \\ \text {Solve}\left [2 \cos (x) \tan \left (\frac {y(x)}{2}\right ) e^{\text {arctanh}(\cos (y(x)))}-\sqrt {\sin ^2(y(x))} \csc \left (\frac {y(x)}{2}\right ) \sec \left (\frac {y(x)}{2}\right ) \left (\log \left (\sec ^2\left (\frac {y(x)}{2}\right )\right )-2 \log \left (\tan \left (\frac {y(x)}{2}\right )\right )\right )&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.857 (sec). Leaf size: 27

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

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (C_{1} e^{- \cos {\left (x \right )}} + 1 \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (C_{1} e^{- \cos {\left (x \right )}} + 1 \right )}\right ] \]