69.6.32 problem 165

Internal problem ID [18075]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 165
Date solved : Thursday, October 02, 2025 at 02:37:21 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (K[1]+1)}dK[1]\&\right ]\left [\int _1^x\cos (K[2])dK[2]+c_1\right ]\\ y(x)&\to -1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.322 (sec). Leaf size: 17

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

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