89.7.18 problem 18

Internal problem ID [24449]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:35:59 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 55

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

\begin{align*} y &= \arccos \left (\frac {\sqrt {3}\, \sqrt {-\sin \left (x \right )^{4}+3 \sin \left (x \right ) c_1}\, \csc \left (x \right )}{3}\right ) \\ y &= \frac {\pi }{2}+\arcsin \left (\frac {\sqrt {3}\, \sqrt {-\sin \left (x \right )^{4}+3 \sin \left (x \right ) c_1}\, \csc \left (x \right )}{3}\right ) \\ \end{align*}

✓ Mathematica. Time used: 21.963 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to -\frac {1}{2} \arccos \left (\frac {1}{6} \csc (x) (-9 \sin (x)+\sin (3 x)-3 c_1)\right )\\ y(x)&\to \frac {1}{2} \arccos \left (\frac {1}{6} \csc (x) (-9 \sin (x)+\sin (3 x)-3 c_1)\right ) \end{align*}

✓ Sympy. Time used: 9.339 (sec). Leaf size: 112

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

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (- \frac {\sqrt {6} \sqrt {\frac {C_{1} - 2 \sin ^{3}{\left (x \right )}}{\sin {\left (x \right )}}}}{6} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (\frac {\sqrt {6} \sqrt {\frac {C_{1} - 2 \sin ^{3}{\left (x \right )}}{\sin {\left (x \right )}}}}{6} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (- \frac {\sqrt {6} \sqrt {\frac {C_{1} - 2 \sin ^{3}{\left (x \right )}}{\sin {\left (x \right )}}}}{6} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {\sqrt {6} \sqrt {\frac {C_{1} - 2 \sin ^{3}{\left (x \right )}}{\sin {\left (x \right )}}}}{6} \right )}\right ] \]