26.5.22 problem Exercise 11.23, page 97

Internal problem ID [6995]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 11, Bernoulli Equations
Problem number : Exercise 11.23, page 97
Date solved : Tuesday, September 30, 2025 at 04:08:03 PM
CAS classification : [_Bernoulli]

\begin{align*} 2 \cos \left (x \right ) y^{\prime }&=y \sin \left (x \right )-y^{3} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.472 (sec). Leaf size: 33

ode:=2*cos(x)*diff(y(x),x) = sin(x)*y(x)-y(x)^3; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {\left (2 \cos \left (x \right )^{2}-1\right ) \left (\cos \left (x \right )-\sin \left (x \right )\right )}}{2 \cos \left (x \right )^{2}-1} \]

✓ Mathematica. Time used: 0.274 (sec). Leaf size: 14

ode=2*Cos[x]*D[y[x],x]==y[x]*Sin[x]-y[x]^3; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{\sqrt {\sin (x)+\cos (x)}} \end{align*}

✓ Sympy. Time used: 1.240 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**3 - y(x)*sin(x) + 2*cos(x)*Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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