80.4.7 problem 7

Internal problem ID [18487]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 31. Problems at page 85
Problem number : 7
Date solved : Monday, March 31, 2025 at 05:37:15 PM
CAS classification : [_Bernoulli]

\begin{align*} y-\cos \left (x \right ) y^{\prime }&=y^{2} \cos \left (x \right ) \left (1-\sin \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=y(x)-cos(x)*diff(y(x),x) = y(x)^2*cos(x)*(1-sin(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.419 (sec). Leaf size: 41

ode=y[x]-Cos[x]*D[y[x],x]==y[x]^2*Cos[x]*(1-Sin[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.096 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1 - sin(x))*y(x)**2*cos(x) + y(x) - cos(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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