68.2.18 problem 23

Internal problem ID [17146]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Review exercises, page 23
Problem number : 23
Date solved : Thursday, October 02, 2025 at 01:45:01 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 12

ode:=diff(y(x),x) = cos(x)^2*sin(x); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {\cos \left (x \right )^{3}}{3}+\frac {1}{3} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 17

ode=D[y[x],x]==Cos[x]^2*Sin[x]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.090 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {1}{3} - \frac {\cos ^{3}{\left (x \right )}}{3} \]