75.3.15 problem 15

Internal problem ID [19919]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 15
Date solved : Thursday, October 02, 2025 at 05:01:16 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.091 (sec). Leaf size: 24

ode:=(exp(y(x))+1)*cos(x)+exp(y(x))*sin(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\ln \left (-\frac {\sin \left (x \right )}{-1+\sin \left (x \right ) {\mathrm e}^{c_1}}\right )-c_1 \]

✓ Mathematica. Time used: 1.218 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to \log \left (-1+e^{c_1} \csc (x)\right )\\ y(x)&\to i \pi \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \log {\left (\frac {C_{1}}{\sin {\left (x \right )}} - 1 \right )} \]