76.4.23 problem 29

Internal problem ID [17346]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 29
Date solved : Monday, March 31, 2025 at 03:55:30 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} {\mathrm e}^{x}+\left ({\mathrm e}^{x} \cot \left (y\right )+2 y \csc \left (y\right )\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 15

ode:=exp(x)+(exp(x)*cot(y(x))+2*y(x)*csc(y(x)))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ {\mathrm e}^{x} \sin \left (y\right )+y^{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 18

ode=Exp[x]+ (Exp[x]*Cot[y[x]]+2*y[x]*Csc[y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [y(x)^2+e^x \sin (y(x))=c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.886 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*y(x)/sin(y(x)) + exp(x)/tan(y(x)))*Derivative(y(x), x) + exp(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + y^{2}{\left (x \right )} + e^{x} \sin {\left (y{\left (x \right )} \right )} = 0 \]