69.1.52 problem 71

Internal problem ID [14134]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 71
Date solved : Monday, March 31, 2025 at 12:10:04 PM
CAS classification : [_Bernoulli]

\begin{align*} y-y^{\prime } \cos \left (x \right )&=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)-diff(y(x),x)*cos(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: 0.439 (sec). Leaf size: 41

ode=y[x]-D[y[x],x]*Cos[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: 2.077 (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)
 

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