88.12.16 problem 18

Internal problem ID [24072]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Miscellaneous Exercises at page 55
Problem number : 18
Date solved : Thursday, October 02, 2025 at 09:56:58 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.090 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.261 (sec). Leaf size: 76

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

\[ \text {Solve}\left [16 \left (-\frac {1}{8} \csc ^2\left (\frac {y(x)}{2}\right )+\frac {1}{8} \sec ^2\left (\frac {y(x)}{2}\right )+\frac {1}{2} \log \left (\sin \left (\frac {y(x)}{2}\right )\right )-\frac {1}{2} \log \left (\cos \left (\frac {y(x)}{2}\right )\right )\right )-16 \left (\frac {x}{2}+\frac {1}{4} \sin (2 x)\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 7.667 (sec). Leaf size: 48

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

\[ - \frac {x}{2} + \frac {\log {\left (\cos {\left (y{\left (x \right )} \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (y{\left (x \right )} \right )} + 1 \right )}}{4} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {\cos {\left (y{\left (x \right )} \right )}}{2 \left (\cos ^{2}{\left (y{\left (x \right )} \right )} - 1\right )} = C_{1} \]