87.4.9 problem 9

Internal problem ID [23275]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 37
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:28:09 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+\frac {y}{\sin \left (x \right )}-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

ode:=diff(y(x),x)+y(x)/sin(x)-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sin \left (x \right )}{\left (\cos \left (x \right )-1\right ) \left (\ln \left (\cos \left (x \right )-1\right )-c_1 \right )} \]

✓ Mathematica. Time used: 0.439 (sec). Leaf size: 54

ode=D[y[x],x]+y[x]/Sin[x]-y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.590 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {\sqrt {\cos {\left (x \right )} + 1}}{\left (C_{1} - \int \frac {\sqrt {\cos {\left (x \right )} + 1}}{\sqrt {\cos {\left (x \right )} - 1}}\, dx\right ) \sqrt {\cos {\left (x \right )} - 1}} \]