68.4.27 problem 27

Internal problem ID [17207]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 27
Date solved : Thursday, October 02, 2025 at 01:53:18 PM
CAS classification : [_separable]

\begin{align*} \frac {\cos \left (y\right ) y^{\prime }}{\left (1-\sin \left (y\right )\right )^{2}}&=\sin \left (x \right )^{3} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 42

ode:=cos(y(x))/(1-sin(y(x)))^2*diff(y(x),x) = sin(x)^3*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arcsin \left (\frac {\cos \left (2 x \right )^{2}+16 c_1 -2 \cos \left (2 x \right )-15}{\cos \left (2 x \right )^{2}+16 c_1 -2 \cos \left (2 x \right )+1}\right ) \]

✓ Mathematica. Time used: 0.579 (sec). Leaf size: 133

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {8 \exp \left (\int _1^{K[2]}\left (\frac {3}{2}+\frac {3}{\cot \left (\frac {K[1]}{2}\right )-1}\right )dK[1]\right ) \left (4 \sin \left (\frac {K[2]}{2}\right )-4 \cos \left (\frac {K[2]}{2}\right ) \cot \left (\frac {K[2]}{2}\right )\right )}{\cot \left (\frac {K[2]}{2}\right )-1}dK[2]-8 \sin ^4(x) \left (\cos \left (\frac {y(x)}{2}\right )-\sin \left (\frac {y(x)}{2}\right )\right )^3 \exp \left (\int _1^{y(x)}\left (\frac {3}{2}+\frac {3}{\cot \left (\frac {K[1]}{2}\right )-1}\right )dK[1]\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.716 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {C_{1} + \sin ^{4}{\left (x \right )} - 4}{C_{1} + \sin ^{4}{\left (x \right )}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {C_{1} + \sin ^{4}{\left (x \right )} - 4}{C_{1} + \sin ^{4}{\left (x \right )}} \right )}\right ] \]