56.7.3 problem Ex 3

Internal problem ID [14107]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 14. Equations reducible to linear equations (Bernoulli). Page 21
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:13:49 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: 0.040 (sec). Leaf size: 14

ode:=sin(y(x))*diff(y(x),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: 0.595 (sec). Leaf size: 81

ode=Sin[y[x]]*D[y[x],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: 0.574 (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)
 

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