89.5.20 problem 20

Internal problem ID [24370]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 43
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:22:23 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} 1-\left (1+2 x \tan \left (y\right )\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 39

ode:=1-(1+2*tan(y(x))*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {2 x \cos \left (2 y\right )-\sin \left (2 y\right )-2 y+c_1 +2 x}{2 \cos \left (2 y\right )+2} = 0 \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 36

ode=1- ( 1+2*x*Tan[y[x]] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [x=\left (\frac {y(x)}{2}+\frac {1}{4} \sin (2 y(x))\right ) \sec ^2(y(x))+c_1 \sec ^2(y(x)),y(x)\right ] \]

✓ Sympy. Time used: 6.220 (sec). Leaf size: 27

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

\[ C_{1} + x \cos ^{2}{\left (y{\left (x \right )} \right )} - \frac {y{\left (x \right )}}{2} - \frac {\sin {\left (y{\left (x \right )} \right )} \cos {\left (y{\left (x \right )} \right )}}{2} = 0 \]