26.5.23 problem Exercise 11.24, page 97

Internal problem ID [6996]
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 11.24, page 97
Date solved : Tuesday, September 30, 2025 at 04:08:08 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\pi }{6} \\ \end{align*}

✓ Maple. Time used: 0.892 (sec). Leaf size: 29

ode:=(x-cos(y(x)))*diff(y(x),x)+tan(y(x)) = 0; 
ic:=[y(1) = 1/6*Pi]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (24 \sin \left (\textit {\_Z} \right ) x -6 \sin \left (2 \textit {\_Z} \right )+2 \pi +3 \sqrt {3}-12 \textit {\_Z} -12\right ) \]

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 62

ode=(x-Cos[y[x]])*D[y[x],x]+Tan[y[x]]==0; 
ic={y[1]==Pi/6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [x=\frac {1}{2} \csc (y(x)) \left (1-2 \int _1^{\frac {\pi }{6}}\frac {1}{2} (\cos (2 K[1])+1)dK[1]\right )+\csc (y(x)) \int _1^{y(x)}\frac {1}{2} (\cos (2 K[1])+1)dK[1],y(x)\right ] \]

✓ Sympy. Time used: 8.067 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - cos(y(x)))*Derivative(y(x), x) + tan(y(x)),0) 
ics = {y(1): pi/6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ x \sin {\left (y{\left (x \right )} \right )} - \frac {y{\left (x \right )}}{2} - \frac {\sin {\left (y{\left (x \right )} \right )} \cos {\left (y{\left (x \right )} \right )}}{2} - \frac {1}{2} + \frac {\sqrt {3}}{8} + \frac {\pi }{12} = 0 \]