87.3.14 problem 14

Internal problem ID [23263]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 26
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:27:23 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

ode:=(1+y(x)^2)*cos(x) = 2*(1+sin(x)^2)*y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {{\mathrm e}^{\arctan \left (\sin \left (x \right )\right )} c_1 -1} \\ y &= -\sqrt {{\mathrm e}^{\arctan \left (\sin \left (x \right )\right )} c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 2.403 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to -\sqrt {-1+e^{\arctan (\sin (x))+2 c_1}}\\ y(x)&\to \sqrt {-1+e^{\arctan (\sin (x))+2 c_1}}\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.656 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} e^{\operatorname {atan}{\left (\sin {\left (x \right )} \right )}} - 1}, \ y{\left (x \right )} = \sqrt {C_{1} e^{\operatorname {atan}{\left (\sin {\left (x \right )} \right )}} - 1}\right ] \]